ORTAÖĞRETİM
MATEMATİK 12 (TEMEL DÜZEY) DERS KİTABI Yazarlar Celal GÜRLER Nusred YILMAZ
1
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Editör
Keziban YILMAZ
Dil Uzmanı Ali Asker Asker OVA OVA
Görsel Tasarım Uzmanı Hakan ÖZ
Ölçme ve Değerlendirme Uzmanı Emrah TOKAT
Program Geliştirme Uzman Yaşar KOCAOĞLU
Rehberlik Uzmanı Yelda Nurdan Nu rdan ÖZDEMİR ÖZDEMİ R TOKAT
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İSTİKLÂL MARŞI Korkma, sönmez bu şafaklarda yüzen al sancak; Sönmeden yurdumun üstünde tüten en son ocak. O benim milletimi milletiminn yıldızıdır, parlayacak; O benimdir, o benim milletimindir milletimindir ancak.
Bastığın yerleri toprak diyerek geçme, tanı: Düşün altındaki binlerce kefensiz yatanı. Sen şehit oğlusun, incitme, yazıktır, atanı: Verme, Verm e, dünyala dünyaları rı alsan alsan da bu cenne cennett vatanı. vatanı.
Çatma, kurban olayım, çehreni ey nazlı hilâl! Kahraman ırkıma bir gül! Ne bu şiddet, bu celâl? Sana olmaz dökülen kanlarımız sonra helâl. Hakkıdır Hakk’a tapan milletimin istiklâl.
Kim bu cennet vatanın uğruna olmaz ki feda? Şüheda fışkıracak toprağı sıksan, şüheda! Cânı, cânânı, bütün varımı alsın da Huda, Etmesin tek vatanımdan beni dünyada cüda.
Ben ezelden beridir hür yaşadım, hür yaşarım. Hangi çılgın bana zincir vuracakmış? Şaşarım! Kükremiş sel gibiyim, bendimi çiğner, aşarım. Yırtarım Yırt arım dağla dağları, rı, enginl enginlere ere sığmam, sığmam, taşarı taşarım. m.
Ruhumun senden İlâhî, şudur ancak emeli: Değmesin mabedimin göğsüne nâmahrem eli. Bu ezanlar -ki şehadetleri dinin temeliEbedî yurdumun üstünde benim inlemeli.
Garbın âfâkını sarmışsa çelik zırhlı duvar, Benim iman dolu göğsüm gibi serhaddim var. Ulusun, korkma! Nasıl böyle bir imanı boğar, Medeniyyet dediğin tek dişi kalmış canavar?
O zaman vecd ile bin secde eder -varsa- taşım, Her cerîhamdan İlâhî, boşanıp kanlı yaşım, Fışkırır ruh-ı mücerret gibi yerden na’şım; O zaman yükselerek arşa değer belki başım.
Arkadaş, yurduma alçakları uğratma sakın; Siper et gövdeni, dursun bu hayâsızca akın. Doğacaktır sana va’dettiği günler Hakk’ın; Kim bilir, belki yarın, belki yarından da yakın.
Dalgalan sen de şafaklar gibi ey şanlı hilâl! Olsun artık dökülen kanlarımın hepsi helâl. Ebediyyen sana yok, ırkıma yok izmihlâl; Hakkıdır hür yaşamış bayrağımın hürriyyet;
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GENÇLİĞE HİTABE Ey Türk gençliği! Birinci vazifen, Türk istiklâlini, Türk Cumhuriyetini, ilelebet muhafaza ve müdafaa etmektir. Mevcudiyetinin ve istikbalinin yegâne temeli budur. Bu temel, senin en kıymetli hazinendir. İstikbalde dahi, seni bu hazineden mahrum etmek isteyecek dâhilî ve hâricî bedhahların olacaktır. Bir gün, istiklâl ve cumhuriyeti müdafaa mecburiyetine düşersen, vazifeye atılmak için, içinde bulunacağın vaziyetin imkân ve şeraitini düşünmeyeceksin! Bu imkân ve şerait, çok namüsait bir mahiyette tezahür edebilir. İstiklâl ve cumhuriyetine kastedecek düşmanlar, bütün dünyada emsali görülmemiş bir galibiyetin mümessili olabilirler. Cebren ve hile ile aziz vatanın bütün kaleleri zapt edilmiş, bütün tersanelerine girilmiş, bütün orduları dağıtılmış ve memleketin her köşesi bilfiil işgal edilmiş olabilir. Bütün bu şeraitten daha elîm ve daha vahim olmak üzere, memleketin dâhilinde iktidara sahip olanlar gaflet ve dalâlet ve hattâ hıyanet içinde bulunabilirler. Hattâ bu iktidar sahipleri şahsî menfaatlerini, müstevlîlerin siyasî emelleriyle tevhit edebi lirler. Millet, fakr u zaruret içinde harap ve bîtap düşmüş olabilir. Ey Türk istikbalinin evlâdı! İşte, bu ahval ve şerait içinde dahi vazifen, Türk istiklâl ve cumhuriyetini kurtarmaktır. Muhtaç olduğun kudret, damarlarındaki asil kanda mevcuttur. Mustafa Kemal Atatürk
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İÇİNDEKİLER ORGANİZASYON ŞEMASI
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8
1. ÜNİTE: GRAFİKLERİN VE TABLOLARIN YORUMLANMAS YORUMLANMASII 1. GRAFİKLERİN VE TABLOLARIN YORUMLANMASI ......... .................. ................. ................. .................. ............. .... 13 1.1. Grafiklerin Grafiklerin ve Tabloların Yorumlanması ..................................... ................................................................... .............................. 13 Ünite Değerlendir Değerlendirme me Testi ................... ...................................... ....................................... ....................................... ..................................... .................. 28
2. ÜNİTE: ÜSTEL FONKSİYONLAR VE UYGULAMALAR UYGULAMALARII 2. ÜSTEL FONKSİY FONKSİYONLAR ONLAR VE UYGULAMALARI ....................................................... 33 2.1. Üstlü İfadelerle Yapılan İşlemlerin Özellikleri ........................................ .......................................................... .................. 33 33 2.2. Üstel Fonksiyo Fonksiyonlar nlar ve Üstel Fonksiyonların Gerçek (Gerçekçi) Hayat Durumlarını Modellemede ve Problem Çözmede Kullanılması .................................................. .................................................. 41 Üstel Fonksiyo Fonksiyonun nun Tanımı ...................................... ............................................................................... ................................................. ........ 42 Üstel Fonksiyon Fonksiyonlar lar ve Üstel Fonksiyo Fonksiyonların nların Gerçek (Gerçekçi) Hayat Durumlarını Modellemede ve Problem Çözmede Kullanılması .................................................. .................................................. 49 Ünite Değerlendirme Testi Testi ...................................... .............................................................................. ........................................................ ................ 55
3. ÜNİTE: ÖLÇME 3. ÖLÇME .......................................................................................................... .................................................................................................................... .......... 59 3.1. Dik Üçgenlerin Gerçek (Gerçekçi) (Gerçekçi) Hayat Hayat Problemlerini Problemlerini Çözmede Kullanılm Kullanılması ası ..... ..... 59 3.2. Üçgenlerin Üçgenlerin Benzerliğinin Gerçek (Gerçekçi) Hayat Durumlarını Modellemede ve Problem Çözmede Kullanılması ......................................................................... ......................................................................... 69 Ünite Değerlendir Değerlendirme me Testi ................... ...................................... ....................................... ....................................... ..................................... .................. 86
4. ÜNİTE: TRİGONOMETRİ VE UYGULAMALAR UYGULAMALARII 4. TRİGONOMETRİ VE UYGULAMALARI ................................................................... ................................................................... 93 4.1. Yönlü Açılar Açılar,, Açı Ölçü Birimlerinden Birimlerinden Derece Derece ve Radyan Radyan ........................................ ........................................ 93
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4.2. Trigonometrik Fonksiyonların Birim Çember Yardımıyla Oluşturu Oluşturulması lması ve Grafikleri .................................... ............................................................................ ............................................................................ .................................... 101 Trigonometrik Fonksiyonların Birim Çember Yardımıyla Oluşturulması ................ 101 Sinüs Fonksiyon Fonksiyonuu ...................................... ............................................................................. ............................................................ ..................... 101 Kosinüs Fonksiyo Fonksiyonu nu ...................................... .............................................................................. ........................................................ ................ 102 Tanjant Fonksiyon Fonksiyonuu ..................................... ............................................................................ ........................................................... .................... 106 İndirgeme Formülleri ................................... .......................................................................... .......................................................... ................... 112 Periyodik Periyod ik Fonksi Fonksiyon yon ................... ....................................... ....................................... ...................................... ..................................... .................. 117 Sinüs Fonksiy Fonksiyonunun onunun Grafiği .................. ..................................... ...................................... ....................................... ......................... ..... 119 Kosinüss Fonksiy Kosinü Fonksiyonunun onunun Grafiği ..................... ........................................... ........................................... .................................. ............. 120 Tanjant Fonksiyonunun Grafiği ......... ................... .................... .................... .................... .................... ................... ................... ............ 120 4.3. Trigonometrik Trigonometrik Fonksiyo Fonksiyonların nların Gerçek (Gerçekçi) Hayat Durumlarını Modellemede ve Problem Çözmede Kullanı Kullanılması lması .................................................. .................................................. 129 Ünite Değerlendir Değerlendirme me Testi ................... ...................................... ....................................... ....................................... ................................... ................ 134 KONU VE ÜNİTE DEĞERLENDİRME SORULARININ CEVAP ANAHTARI ................................ 137 SÖZLÜK ................. .............................. ........................... ........................... .......................... ........................... ........................... .......................... ........................... ......................... ........... 139 139 KAYNAKÇA KA YNAKÇA ......................... .................................................... ...................................................... ...................................................... .................................................. ....................... 142 GÖRSEL KAYNAKÇA ........... ...................... ...................... ..................... ..................... ...................... ...................... ...................... ...................... ...................... ................. ...... 142 SEMBOL VE GÖSTERİMLER ............... ............................... ................................ ................................ ................................ ................................ ....................... ....... 143
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ORGANİZASYON ŞEMASI
Bu sayfada ünite numarası ve ünitenin konusu yer almaktadır.
Bu başlık başlık altın altında, da, ünitede yer alan konular verilmiştir.
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Bu başlık başlık altın altında, da, ünite numa ralarına ve ünite isimlerine yer verilmiştir.
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Bu başlık başlık altın altında, da, her konu nun sonunda öğrenilen bilgile rin pekiştirilmesi için sorulara yer veril ve rilmiş miştir. tir.
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Bu başlık başlık altın altında, da, bölüm için de yer alan kavramların tanımlan masına yer verilmiştir.
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Bu başlık başlık altın altında, da, konuyu anla mamıza yardımcı olacak örnek ve çözümlere yer verilmiştir.
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Bu başlık başlık altın altında, da, konuyla ilgili öğrenilmesi gereken bilgilere yer verilmiştir.
Bu başlık başlık altın altında, da, konuyla ilgili kural, formül ve bağıntılara yer verilmiştir.
Bu başlık başlık altın altında, da, konuyla ilgili uyarıların yapıldığı ve konunun önemli noktalarına dikkatin çekil diği bilgilere yer verilmiştir.
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Etkinlik
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SAYILAR VE CEBİR
1.ÜNİTE:
GRAFİKLERİN GRAFİKLE RİN VE TABLOLARIN YORUMLANMASI
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Ünitenin Konuları 1. GRAFİKLERİN VE TABLOLARIN YORUMLANMASI 1.1. Grafiklerin ve Tabloların Yorumlanması
Neler Öğreneceğiz?
Grafiklerin Grafikler in ve Tabloların Yorumlanması konusunda; • Bir grafiğin veya tablonun yorumlanmasını gerektiren problemleri çözmeyi öğreneceğiz.
Ünitenin Kavram ve Terimleri Terimler: Grafik, tablo,
değişim oranı (hızı)
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1. GRAFİKLERİN VE TABLOLARIN YORUMLANMASI
1. GRAFİKLERİN VE TABLOLARIN YORUMLANMASI Düşünelim Karmaşık verileri daha iyi analiz edebilmek için verileri görsel h âle getirmek iyi bir çözümdür. Grafiklerin kullanımı, verileri görsel h âle getirip anlaşılmasını kolaylaştırdığı gibi fonksiyonların özelliklerini kullanarak eldeki verilerden yeni bilgilere ulaşmamıza da imk ân sağlar. Günlük yaşantımızda grafikler ve tablolar; gazetelerde, gazetelerde, dergilerde, televizyon programlarında, bro şürlerde, sınav sonuç belgelerinde, internet gibi farklı ortamlarda verileri özetlemek için kullanılır. Örneğin; • Bir ay süresince altın ve petrol fiyatlarındaki değişikliği, • Bir çocuğun yıllara göre değişen kilosunu, • İki ülke arasındaki nüfus artışının kıyaslanmasını, • Bir şirketin gelir ve gider durumlarını, • Farklı iki şehirde yaşayan kişilerin ödediği su faturalarının kıyaslanmasını, • Öğrencilerin sınavlarda sınavlar da almış olduğu notlar gibi verileri daha etkili bir şekilde özetlemede ve kar şılaştırmada grafikleri çok sık kullanırız.
1.1. Grafiklerin ve Tabloların Yorumlanması Örnek
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1. Grafklerin ve Tabloların Yorumlanması Çözüm Yıllar
1970
1980
1990
2000
2010
2014
Artış hızı (binde) (binde)
25,19
20,65
21,71
18,28
15,9
13,3
Tablo incelendiğinde yıllar geçtikçe nüfus artış hızının azaldığı görülmektedir. Fakat 19801990 arasında nüfus artış hızının arttığı söylenebilir.
Örnek Ülkemizde 2006 'dan 2015 'e kadar olan yılların Mayıs ayına ait altın fiyatlarını tablo ve grafik yardı mıyla gösterelim.
Çözüm Yıllar
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
Altın Fiyatı (TL) (TL)
31
28
35
46
60
76
92
83
86
101
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1. Grafklerin ve Tabloların Yorumlanması Grafiğe bakarak kolay bir şekilde yıllar geçtikçe altın fiyatlarının nasıl değiştiğini görebiliriz. Örne ğin altın fiyatlarının genel eğiliminin yıllar geçtikçe artmakta olmasıdır. Ancak 2007 ve 2013 yılların da altın fiyatlarının düştüğü de görülmektedir. Ayrıca, 201 Ayrıca, 20111 yılı yılında ndan n 201 20122 yılı yılına na geç geçildi ildiğin ğinde de altı altın n fiya fiyatlar tlarının ının cid ciddi di bir artı artışş gös gösterd terdiğin iğinii gör görmek mekteyiz.
Örnek Ayşe Hanım 3 kg olarak doğan çocuğu Ahmet'in yıllara göre değişen kilosuyla ilgili olarak aşağıdaki tabloyu oluşturmuştur.
Yıllar
2007
2008
2009
2010
2011
2012
2013
2014
2015
Kilo (kg)
3
8
12
18
20
24
29
32
36
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1. Grafklerin ve Tabloların Yorumlanması Grafiğe bakılarak kolay bir şekilde yıllar geçtikçe Ahmet'in kilosunun artmakta olduğu görü lür. Ancak 2010 yılından 2011 yılına geçişte kilosunun diğer yıllara göre daha az arttığı da söy lenebilir. Diğer taraftan 2009 yılından 2010 yılına geçişte ise kilosu diğer yıllara göre en çok ar tışı göstermiştir.
Örnek Bir otopark işletmesinin, araçların kaldık ları süre t saat olmak üzere, otopark ücretini f ^ t h = 5 + ^ t − 1h $ 3 doğrusal fonksiyonuna göre hesapladığını varsayalım. Bu fonksiyonun grağini çizip yorumlayalım yorumlayalım..
Çözüm t = 1 saat için müşterinin ödediği ücret, f ^1 h = 5 + ^1 − 1h $ 3
&
f ^1 h = 5 TL
t = 2 saat için müşterinin ödediği ücret, f ^2 h = 5 + ^2 − 1h $ 3
&
f ^2h = 8 TL
t = 3 saat için müşterinin ödediği ücret,
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1. Grafklerin ve Tabloların Yorumlanması
Örnek Dünya Bankası Dünya Gelişim göstergelerine gö re Türkiye ve Almanya'nın 2004 ve 2014 yıllarına ait nüfusları aşağıdaki tabloda verilmiştir. Yıllar Ülkeler Türkiye Almanya
2004
2014
66.845.635 82.516.260
77.695.904 81.619.000
(www.tuik.gov.tr )
Buna göre 2004 - 2014 yılları arasında her iki ül kenin nüfuslarındaki değişimi bularak yorumlayalım.
Çözüm
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1. Grafklerin ve Tabloların Yorumlanması Grafikleri inceleyerek 2004 - 2014 yılları arasındak arasındakii nüfus artışının ve azalışının ortalama ola rak bu oranlarda olduğunu varsayarak aradaki yıllar için yaklaşık değerlendirmeler yapabiliriz. Bu değerlendirmeler değerlendirmeler için ilgili doğrunu doğrunun n (x yıl, y nüfus) bilgilerini kullanırız. Ancak bu aralığın dı şında kalan geçmiş veya gelecek yıllardaki nüfus artışı oranlarını bilmediğimizden yapacağımız varsayımlar bize doğru sonuçlar vermeyebilir.
Örnek Bir firmanın firmanın zamana (x yıl) bağlı bağlı gelir gelir (milyon (milyon TL) fonksiyonu f ^x h = 2x − 6 , gider fonksiyonu g ^ x h = x + 2 olarak veriliyor. Buna göre f(x) ve g(x) fonksiyonlarının; a. Graklerini aynı düzlemde çizelim.
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1. Grafklerin ve Tabloların Yorumlanması
Örnek A şehrindeki şehrindeki Ayşe Hanım ile B şehrinde şehrindeki ki Zeynep Zeynep Hanım'ın aylık su faturasının hesaplanması yandaki doğrusal grafikte verilmiştir. A ve B şehirlerinden hangisinin hangi durumlar da daha ekonomik olduğuna nasıl karar verirsiniz? Verilen iki ücret tarifesine göre ödenecek su fatu rasının harcanan su miktarına bağlı olarak değişimi ni karşılaştırarak aşağıdaki soruları cevaplayalım cevaplayalım..
denen ücret (TL) 0
A B
0
0
0
a. Aylık
20 ton su tüketen bir kişi hangi şehirde yaşasaydı daha az ücret öderdi? b.
Aylık su tüketimi kaç ton olduğunda her iki şe -
Su (ton) (ton) 0 5
10
15
20
25
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1. Grafklerin ve Tabloların Yorumlanması
Örnek A şehrinde taksimetre ücretleri açılış 8 TL sabit ücret ve her bir kilometre için 0,4 TL olarak belirlenmiştir.. B şehrinde taksimetre ücretleri açılış 6 TL samiştir bit ücret ve her bir kilometre için 0,6 TL olarak belirlenmiştir. Buna göre bu iki şehir için; a. Taksimetre
ücretlerinin maliyet denklemlerini
yazalım. b. A ve
B şehirlerindeki taksimetre ücretlerini kıyaslayarak yorumlayalım.
Çözüm a. A şehrindeki taksimetre ücretinin maliyeti M A ve B şehrindeki taksimetre ücretinin mali -
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1. Grafklerin ve Tabloların Yorumlanması Verilen tablo ve doğrusal grak incelendiğinde 10 km yol gidildiğinde her iki şehirde de aynı ücretin ödeneceği kolayca görülebilir. 0 - 10 km arası yol gidildiğinde B şehrinde ödenen ücretin daha az (ekonomik) olduğu gö rülmektedir. Aynı şekilde 10 km den sonra ise A şehrindeki ödenen ücretin daha az (ekonomik) olduğu görülmektedir.
Örnek Bir yayınevi yayınladığı kitapların tanesini 18 TL den satmayı planlamaktadır. Kitabın başlangıçtaki maliyeti 6.000 TL olup kitap başına birim maliyeti 8 TL dir. Buna göre; a. Maliyet (M) ve gelir
(G) denklemlerini yazalım.
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1. Grafklerin ve Tabloların Yorumlanması c. Gelir
(G) denkleminden maliyet (M) denklemi çıkarılınca kâr denklemi elde edilir.
K = G−M
&
K = 18x − ^6000 + 8x h
&
K = 10x − 6000
olur.
1000 adet kitap satılırsa elde edilen kâr, K = 10 $ 10 1000 − 6000 & K = 4000 TL dir.
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1. Grafklerin ve Tabloların Yorumlanması
Etkinlik •
Aşağıda verilen çizgi grafğinde x ekseni geçen süreyi göstermektedi göstermektedir. r.
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1. Grafklerin ve Tabloların Yorumlanması Çocuk nüfusun toplam nüfus içindeki oranı (1935 – 2075) 60
50
40
%
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1. Grafklerin ve Tabloların Yorumlanması ^x1, y1h değerlerinden ^x2, y2h değerlerine geçişte yaşanan değişim oranını grafik olarak yandaki gi bi gösterebiliriz.
Hatırlayacağınız gibi bir doğrunun eğimi, doğru üzerindeki herhangi iki nokta arasındaki dikey deği şimin yatay değişime oranı olarak tanımlanmaktay
y y = f(x) y2
(x2, y2)
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1. Grafklerin ve Tabloların Yorumlanması
x
g(x)
x değerindeki değişim
g(x) değerindeki değişim
Değişim oranı (hızı)
^ x2 − x 1h
^g^x 2 h − g ^x1hh
g ^ x 2 h − g ^ x1 h
− 12 − ^− 17h = 5
-3
-17
− 2 − ^− 3 h = 1
-2
-12
0 − ^− 2 h = 2
− 2 − ^− 12h = 10
x2 − x1 5/1 = 5 10/2 = 5
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3.Ölçme b) 5 – 12 – 13 üçge üç geni, ni, k ∈ R + olmak üzere yandaki dik üçgenden; k = 1 ⇒ 5 – 12 – 13,
k = 2 ⇒ 10 – 24 – 26
dik üçgenleri elde edilir.
c) 8 – 15 15 – 17 üç üçge geni, ni,
d) 7 – 24 – 25 üç üçge geni, ni,
k = 1 ⇒ 8 – 15 – 17 üçgeni bulunur bulunur..
k = 1 ⇒ 7 – 24 – 25 üçgeni bulunur.
Örnek Yandaki Y andaki şekilde ABC bir üçgen, ADC dik üçgen, [AE] ⊥ [BC]
|DC| = 24 cm, |EC| = 20 cm, |AB| = 17 cm ve |BE| = 8 cm olduğuna göre |AD| = x kaç santimetredir? Bulalım.
Çözüm
AEB dik üçge üçgeni, ni, dik kenarı 8 cm ve hipotenüsü 17 cm olduğundan kenarları tam sayı olan 8 – 15 – 17 dik üçgenidir üçgenidir.. O zaman |AE| = 15 cm olur. AEC dik üçge üçgeni, ni, dik kenarları 15 cm ve 20 cm olduğundan kenarları tam sayı olan 15 – 20 – 25 dik üçgenidir.. Buradan |AC| = 25 cm bulunur. ADC dik üçge dir üçgeni ni ise dik kenarı 24 cm ve hipotenüsü 25 cm olduğundan kenarları tam sayı
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3. Ölçme Örnek
Evden eve taşımacılık yapan bir nakliye firması taşıma işlerinde asansör kullanmaktadır. Resimdeki binanın 36 metre yüksekliğindeki 10. katında bulunan bir daireden eşyaları almak için asansör apartmana en fazla 27 metre yanaşabilmektedir. Buna göre daireye ulaşabilmesi için asansörün kaç metre olması gerekmektedir? Bulalım.
Çözüm
Bu problemi yandaki ABC dik üçgeni ile modelleyelim. [AB] ⊥ [BC] olduğundan dik üçgende Pisagor Teoremi Teoremi ile 2 2 2 |AC| = |AB| + |BC| |AC|2 = 362 + 272 |AC| = 45 metre bulunur. Sonuçta asansörün daireye ulaşabilmesi için 45 m olması gerekmektedir.
Örnek Yandaki ABC üçgeninde [AD] ⊥ [BC], |AC| = 17 cm, Yandaki |AB| = 10 cm ve |BD| = 6 cm olduğuna göre |DC| = x kaç santimetredir? Bulalım.
Çözüm
ADB dik üçge üçgeni, ni, dik kenarı 6 cm ve hipotenüsü 10 cm olduğundan kenarları tam sayı olan 6 – 8 – 10 dik üçgenidir. Buradan |AD| = 8 cm bulunur bulunur.. ADC dik üç üçge geni ni ise dik kenarı 8 cm ve hi-
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3.Ölçme Örnek GeoGebra matematik / geometri yazılımını kullanarak kenar uzunlukları 3 birim, 4 birim ve 5 birim olan bir üçgenin dik üçgen olduğunu görelim. Dinamik matematik / geometri yazılımını açalım ve “geometri” sekmesini tıklayalım. Yazılımın üst kısmındaki da ki 3. ku kutu tuda da bu bulu luna nann ( ) “ve “veri ri-len uzunlukta doğru parçası” aracı ile |AB| = 5 br ve 6. kutuda bulunan ( ) “merkez ve yarıçapla çember” aracı ile yarıçapı 3 br ve 4 br olan A ve B merkezli çemberleri çizelim. Yazılımın üst kısmındaki 5. kutuda buluna bulunann ( ) "çokgen" "çokgen" aracı aracını nı kullanara kullanarakk ABC üçgeni üçgenini ni çizelim çizelim.. Yazılı Yazılımın mın üst üst kısmınkısmındaki da ki 8. kut kutud udaa bul bulun unan an (
ë = 90° ölçülür. Görüldüğü gibi kenar uzunluk) “aç “açı” ı” ara aracı cıyl ylaa m(B m(B AC)
ları 3 birim, 4 birim ve 5 birim olan bir üçgen dik üçgen olmaktadır.
Bunları Bilelim
Öklit Bağıntıları
Bir ABC dik üçgeninde m( A) ABC dik ë = 90° olmak üzere ABC üçgeninin hipotenüsüne ait yükseklik h, yüksekliğin hipotenüs üzerinde ayırdığı parçaların uzunlukları p ve k olsun. Buna göre;
2
♦ h
♦
= p.k (yükseklik bağıntısı)
c2 = p.a, b2 = k.a (dik kenar bağıntıları)
♦ a
. h = b . c (alan bağıntısı)
bağıntıları geçerlidir. Bu bağıntılar Öklit Bağıntıları olarak adlandırılır adlandırılır..
Örnek Yandaki ABC dik üçgeninde [AB] ⊥ [AC], [AH] ⊥ [BC], Yandaki |AH| = 6 cm ve |BH| = 4 cm olduğuna göre |HC| = x kaç santimetredir? Bulalım.
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3. Ölçme Örnek Yandaki ABC dik üçgeninde [AH] ⊥ [BC], |BH| Yandaki |BH| = 16 cm ve |HC| = 9 cm olduğuna göre h, b ve c uzunlukları kaçar santimetredir? Bulalım.
Çözüm Öklit yükseklik bağıntısından
h2 = p.k ⇒ h2 = 16 . 9 ⇒ h = 12 cm bulunur.
Öklit dik kenar bağıntılarından
b2 = k.a ⇒ b2 = 9 . 25 ⇒ b = 15 cm bulunur.
Öklit dik kenar bağıntılarından c2 = p.a ⇒ c2 = 16 . 25 ⇒ c = 20 cm bulunur.
Örnek
Yandaki haritada B şehrinden D şehrine hareket Yandaki eden bir uçağın konumu gösterilmiştir gösterilmiştir.. Uçak havada C şehri üzerindeki A noktasına geldiğinde B AD olmaktadır. B ve C şehirşehirë açısının ölçüsü 90° olmaktadır. leri arası uzaklık 90 km, C ve D şehirleri arası uzaklık 40 km olduğuna göre uçak C şehri üzerindeyken uçağın yerden yüksekliği kaç km 'dir? Bulalım.
Çözüm
Bu problemi yandaki ABD dik üçgeni ile modelleyelim. ABD dik üçgeninde [AC] ⊥ [BD] olduğundan Öklit yükseklik bağıntısı ile; 2 ⇒ |AC| = |BC|.|CD| 2 ⇒ h = 90.40 = 3600
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3.Ölçme Örnek Yandaki resimde Yandaki r esimde bir çiftçi bahçesine meyve fidanları dikmektedir. A noktasında bulunan çiftçi fidanı diktikten sonra bahçesinin yanından geçen nehrin bir E noktasından su alarak D noktasındaki meyve ağacını sulamak istemektedir. Çiftçinin nehire uzaklığı 9 m, ağacın nehire uzaklığı 7 m ve |BC| = 12 m dir. Buna göre çiftçinin nehirden suyu alıp ağacı sulamaya giderken katedebileceği en kısa |AE| + |ED| uzaklığı kaç metredir? Bulalım.
Çözüm
Bu problemi yandaki gibi birbirine dik olan [AB], [BC] ve [CD] ile modelleyelim. [AB] ve [CD] nı 7 m uzatalım ve FGCB dikdörtgenini oluşturalım. A ve G noktalarını birleştirdiğimizde en kısa |AE| + |ED| uzaklığı |AG| uzunluğuna eşittir eşittir.. AFG dik üçgeninde Pisagor Teoremi Teoremi ile; 2 2 2 ⇒ |AG| = |AF| + |FG| 2 2 2 ⇒ |AG| = 16 + 12 ⇒ |AG| = 20 m bulunur bu lunur.. Sonuçta en kısa |AE| + |ED| uzaklığı 20 m bulunur.
Örnek
Yandaki krokide Mustafa A noktasındaki evinden Yandaki işaretli sokakları kullanarak D noktasındaki okuluna yürüyerek gidip gelmektedir. |AB| = 50 m, |BC| = 120 m ve |CD| = 40 m dir. Buna göre Mustafa 'nın evi ile okulu arasındaki mesafe kuş uçuşu kaç metredir? Bulalım.
Çözüm
Bu problemi yandaki gibi birbirine dik olan [AB], [BC] ve [CD] ile modelleyelim. [AB] nı uzatarak BCDE dikdörtgenini oluşturalım. Bu modelde kuş uçuşu mesafe |AD| uzaklığıdır uzaklığıdır.. AED dik üçgeninde Pisagor Teoremi Teoremi ile;
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Konu Değerlendirme Soruları-1 ABC dik üçgen [AB] ⊥ [AC]
1.
4.
|AB| = 4 cm |AC| = 8 cm
ABC bir dik üçgen, [AB] ⊥ [AC], |AB| = x – 1 cm, |AC| = 2x cm, |DC| = 3x – 5 cm
Yukarıdaki verilere göre x kaç santimetred santimetredir? ir? 5 B) 5 C) 8ñ 3 D) 10 E) 12 A) 4ñ
Yukarıdaki verilere göre x kaç santimetred santimetredir? ir? A) 2
2.
ABC bir üçgen [AD] ⊥ [BC] |AB| = 5 cm |BD| = 1 cm |DC| = 5 cm
5.
B) 3
C) 4
D) 5
E) 6
Yukarıda ukarıdaki ki verilere göre x kaç santimetredir santimetredir?? 2 C) 7 D) 7ñ 2 E) 8 A) 6 B) 6ñ
ADE bir üçgen, ABC bir dik üçgen, [AB] ⊥ [BC], [AE] ⊥ [DC], |AB| = 4 cm, |BC| = 3 cm, |CE| = 16 cm, |DE| = 20 cm Yukarıdaki verilere göre x kaç santimetredi santimetredir? r?
3.
A) 15
B) 10
C) 16
D) 12
E) 13
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Konu Değerlendirme Soruları-1 7.
[AB] ⊥ [BC] [BC] ⊥ [CD] |BC| = 6 cm |AB| = 5 cm |CD| = 3 cm
10.
ABC ve ABD birer dik üçgen, [AD] ⊥ [BC], [AB] ⊥ [BD], [AB] ⊥ [AC], |BH| = 2 cm, |HD| = 1 cm
Yukarıdaki verilere göre |AD| kaç santimet Yukarıdaki redir? A) 6 B) 8 C) 10 D) 12 E) 16
ABC dik üçgen [AH] ⊥ [BC] [AB] ⊥ [AC] |BH| = 4 cm |HC| = 16 cm
Yukarıdaki verilere göre x kaç santimetredi santimetredir? r? A) 4 B) 6 C) 8 D) 10 E) 12
11.
ABD ve BEC dik üçgen
[AB] ⊥ [BD]
[BE] ⊥ [AC] |AE| = 12 cm |ED| = 3 cm |CD| = 5 cm
Yukarıdaki verilere göre x kaç santimetredi Yukarıdaki santimetredir? r? A) 6 B) 8 C) 9 D) 10 E) 12
8.
Yukarıdaki verilere göre x kaç santimetredir Yukarıdaki santimetredir?? A) 8 B) 10 C) 12 D) 16 E) 20
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3. Ölçme
3.2. Üçgenlerin Benzerliğinin Gerçek (Gerçekçi) Hayat Durumlarını Modellemede ve Problem Çözmede Kullanılması Ölçme problemlerinin çözümünde kullanılan kavramlardan birisi de benzerliktir benzerliktir.. Gerçek hayatta ölçülmesi zor olan uzunluk, alan ve hacim problemleri pro blemleri benzer üçgenlerle modellenerek çözüm yapılabilir.. Modellemede kullanılan benzerlik teoremlerini uygulamalarıyla inceleyelim. labilir
Tanım
Benzerlik Kavramı
İki üçgen arasında yapılan bire bir eşlemede karşılıklı açılar eş ise bu iki üçgen benzerdir benzerdir.. m( A) ë = m(ë D), m(ë B) = m(ëE), m(ë C) = m(ë F)
¿ BC ∼ D ¿ EF olur. ⇒ A
Benzer üçgenler yazılırken harf sıralaması sıralaması açıların açıların eşitliğine göre yazılır yazılır.. (ABC
DEF
olur.)
Benzer üçgenlerde eşit açıların karşısındaki kenarlar orantılıdır. Bu orantıdan elde edilen orana benzerlik oranı denir ve k ile gösterilir. k = 1 olursa benzer üçgenler eş üçgenler olur. Üçgenle ≅ D ¿ EF şeklinde ifade edilir. rin eşliği A ¿ BC
Örnek
∼ A ¿ ED ve m(Bë DE) = 110° olduğuna göre ABC üçgeninde A ¿ BC = a kaç derecedir? Bulalım. m(Aë CB)
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3. Ölçme Bunları Bilelim
Açı-Açı Benzerlik Teoremi Teoremi
İkişer açısı eş olan ola n üçgenler benzerdirler. (Bu durumda 3. açılar da eş olur.) m(A)=m(D)
∼ D ¿ EF olur. ⇒ A ¿ BC
m(B)=m(E)
Sonuçta m(ë C) = m(ë F) olur. Karşılıklı kenarlar
orantılı olduğundan
a d
=
b e
=
c
=k
f
yazılabilir.
Örnek
= m(ë D), m(ë C) = m(ë E), |AB| = 18 cm, Yukarıdaki ABC ve FDE üçgenlerinde m(ë B) Yukarıdaki |BC| = 27 cm ve |FD| = 6 cm olduğuna göre |DE| = x kaç santimetredir? Bulalım.
Çözüm = m(ëD) ve m(ë C) = m(ë E) oldu ABC ve FDE üçgenlerinde m(ë B) olduğu ğu için Açı-Açı Benzerlik Teore ∼ F ¿ ¿ DE olur. Buradan m( A) ë = m(ëF) olur. mi’ nden A ¿ BC
6 18
=
x 27
⇒ x = 9 cm bulunur.
Örnek
= m(Aë BC), Yandaki ABC ve AED üçgenlerinde m(A Yandaki m(Aë ED) |AD| = 8 cm, |AE| = 6 cm, |DE| = 7 cm ve |DB| = 4 cm olduğuna göre |BC| = x kaç santimetredir? Bulalım.
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3. Ölçme Örnek
Bir araştırmacı Mısır 'da karşılaştığı bir piramitin yüksekliğini hesaplamak istemektedir. Boyu 1,7 m olan araştırmacının gölgesinin uzunluğu 2 m olduğu anda piramitin gölgesi 40 m olduğuna göre araştırmacı piramitin boyunu kaç metre olarak hesaplamıştır? Bulalım.
Çözüm Bu problemi aşağıdaki gibi ABC ve DEF dik üçgenleriyle modelleyelim.
Güneş ışınları yer düzlemiyle her noktada aynı açıyı yaptığı için m(Aë BC) = m(DëEF) dir. = m(DëFE) = 90° olduğundan Açı-Açı Benzerlik Teoremi'nden ∼ D ¿ EF bulunur. m(Aë CB) Teoremi'nden A ¿ BC BC
EF
=
AC
DF
2 40
=
1,7 h
2.h = 40.1,7
h = 34 m bulunur.
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3. Ölçme Örnek Dinamik matematik / geometri yazılımını kullanarak kenar uzunlukları |AC|= 5 br br,, |BC|= 7 br ve ∼ D ¿ EF), |FD| = 10 br, |AB|= 8 br olan ABC üçgenine benzer olan bir DEF üçgeni için (A ¿ BC |FE| = 14 br ve |DE| = 16 br olduğunda benzer üçgenlerin iç açılarının eş olduğunu görelim. Dinamik matematik / geometri yazılımını açalım ve “geometri” sekmesini tıklayalım. Yazılımın üst kısmındaki da ki 3. kut kutud udaa bul bulun unan an ( ) “ve “veri ri-len uzunlukta doğru parçası” aracı ile |AB| = 8 br çizelim. Yazılımın üst kısmındaki 6. kutuda bulunan ( ) “merkez ve yarıçapla çember” aracı ile yarıçapı 5 br ve 7 br olan A ve B merkezli çemberleri çizelim. Yazılımın üst üst kısmın kısmındak dakii 5. kutuda kutuda bulunan bulunan ( ) "çokgen "çokgen"" aracını aracını kullan kullanarak arak ABC ABC üçgeni üçgenini ni çizeli çizelim. m. Benzer yollardan kenarları 10 br, 14 br, 16 br olacak şekilde DEF üçgenini çizelim. Sonra yazılımın üst kısmınd kısmındaki aki 8. 8. kutuda kutuda bulunan bulunan ( ) “açı” “açı” aracıy aracıyla la ABC ABC ve DEF üçgenl üçgenlerini erininn iç açılar açılarının ının ölçülerini hesaplayalım. Görüldüğü gibi benzer olan iki üçgenin iç açılarının ölçüleri karşılıklı olarak eştir.
Örnek
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3. Ölçme Örnek Pikniğe giden Cemal doğada gördüğü bir ağacın boyunu ölçmek istemiştir. istemiştir. Bunun için yan daki resimde görüldüğü gibi 20 cm uzunluğundaki bir cetveli ağaçtan 25 m mesafede, göz hizasından 50 cm uzakta yere dik olacak şekilde, cetvelin iki ucundan ağacın kökünü ve te pesini görecek şekilde tutmuştur. Buna göre Cemal ağacın boyunu kaç metre ölçmüştür? Bulalım.
Çözüm Bu problemi aşağıdaki gibi ABC ve ADE üçgenleri ile modelleyelim. ABC üçgeninde [DE] // [BC] olduğundan ∼ A ¿ DE Temel Orantı Teoremi 'ne göre A ¿ BC = m(Aë DE) ve olur.. Bu durumda m(Aë BC) olur m(Aë CB) = m(Aë ED) dir.
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3. Ölçme Örnek
Yandaki resimde verilen koni tabanlı siloda |TB| Yandaki |TB| = 1 m ve |BD| = 2 m dir. Silonun içinde AB seviyesine kadar 8 m3 tahıl vardır. Buna göre siloyu CD seviyesine kadar doldurmak için üzerine kaç metreküp tahıl eklenmelidir? Bulalım.
Çözüm
Bu problemi yandaki (T (T,, CD) dairesel dik konisi ile modelleyelim. (T (T,, AB) ve (T (T,, CD) dairesel dik konileri konile ri benzerdir. &
k=
TB TD
&
k=
1 3
bulunur. Benzer konilerin hacimleri oranı
benzerlik oranının küpüne k üpüne eşittir.
&
&
Örnek
V(T,AB) = k3 & V(T,CD)
V(T,AB) 3 8 = 1 & = 1 3 27 V(T,CD) V(T,CD)
V(T,CD) = 216 m3 olur.
a k
Buradan eklenecek tahıl miktarı
216 – 8 = 208 m 3 bu bulunur.
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3. Ölçme Örnek
Yukarıdaki resimde bir öğrenci uzunluğunu 21,5 m ölçtüğü bir ağaçtan yararlanarak karşıdaki ka Yukarıdaki yalığın yüksekliğini hesaplamak istemektedir. Öğrenci ağacın kayalığa uzaklığını 250 m ölçmüş ve kendisi de ağacın ucundan kayalığın zirvesini görecek şekilde ağaçtan 50 m uzaklıkta pozisyon almıştır. Öğrencinin göz hizasının yerden yüksekliği 1,5 m olduğuna göre öğrenci kayalığın yüksekliğini kaç metre hesaplamıştır? Bulalım.
Çözüm
Bu problemi şekildeki ABC dik üçgeni ile modelleyelim. Göz hizasının yerden yüksekliği 1,5 m olduğundan dolayı |DE| = 21,5 – 1,5 = 20 m ve
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3. Ölçme Örnek
Yandaki şekilde [AB] // [DE], [AE] [AE ] ∩ [BD] = {C}, |AC| = 6 cm, |BC| = 9 cm ve |CE| = 8 cm olduğuna göre |CD| = x kaç santimetredir? Bulalım.
[AB] // [DE] ve [AE] ∩ [BD] = {C} olduğundan Kelebek Benzerliği'nden A ¿ BC açıların ın eşitliğinden, ∼ C ¿ DE dir. İç ters açılar ë = m(ë E) ve m(ë B) = m(ëD) dir. m( A)
Çözüm
⇒
6 8
=
9 x
⇒ x = 12 cm bulunur.
Örnek Yandaki Y andaki resimde bir kum saati verilmiştir. Kum saatinin üst ü st bölme-
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3. Ölçme Bunları Bilelim
II.Tales Benzerlik Teoremi
Yandaki şekilde d1 // d2 // d3 ve bu doğruların iki keseni d4 ve d5 Yandaki olmak üzere; a, b, c, d ve x, y, z uzunlukları arasında aşağıdaki bağıntılar geçerlidir:
1)
2)
a b
=
c
y x z y
d
−
−
=
a b
Örnek
Yandaki şekilde [DC] // [EF] // [AB], Yandaki [AB], |DC| = 20 cm, cm, |EF| = 24 cm, |DE| = 5 cm ve |EA| = 10 cm olduğuna göre |AB| = x kaç santimetredir? Bulalım.
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3. Ölçme Bunları Bilelim
Kenar-Açı-Kenar Benzerlik Teoremi
Birer açısı eş ve bu açıyı a çıyı oluşturan kenarları orantılı or antılı olan üçgenler benzerdirler.
= m(ë E), |EF| = k.|BC| ve Yandak andakii şekild şekildee m(ë B) ∼ D ¿ EF olur. |DE| = k.|AN| olduğunda A ¿ BC
Buna göre
EF BC
=
DE AB
=k
DF
= k olur.
AC
Örnek
Yandaki ABC üçgeninde |AD| = 7 cm, Yandaki cm, |AE| = 5 cm, |DB| = 3 cm, |EC| = 9 cm ve |DE| = 6 cm olduğuna göre
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3. Ölçme Önemli Bilgi
Benzerlik-Yardımcı Benzerlik-Y ardımcı Eleman İlişkisi
Benzer iki üçgenin benzerlik oranı k olmak üzere, yardımcı elemanlarının (yükseklik, açıortay, kenarortay) ve çevrelerinin oranı benzerlik oranına o ranına (k) eşittir.
∼ D ¿ EF olmak üzere, Yandak andakii şekild şekildee A ¿ ¿ BC
ha hd
Önemli Bilgi
=
n A nD
=
Va Vd
=
a d
&
=
Çevre ( ABC) &
Çevre (DEF)
Benzerlik-Alan İlişkisi
Benzer üçgenlerin alanları oranı benzerlik oranının karesine eşittir.
=
k olur.
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3. Ölçme Örnek
Bir çiftçi, alanı 1600 m2 olan üçgensel bölge şeklindeki tarlasını, iki sınırının ortası olan K ve L noktalarından KL doğrusu boyunca bölerek iki oğluna paylaştırıyor.. Küçük oğluna üçgensel bölge olan tarlayı, laştırıyor büyük oğluna da dörtgensel bölge olan tarlayı veriyor.. Buna göre çiftçinin her bir oğluna düşen tarla yor miktarlarını bulalım.
Çözüm
Bu problemi yandaki ABC üçgeni ile modelleyelim. K ve L orta noktalar olduğu için [KL] orta tabandır ve [KL] // [BC] dir. Temel ∼ A ¿ BC olur. Orantı Teoremi 'nden A ¿ KL
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3. Ölçme Önemli Bilgi
Yandaki şekilde eşit aralıklı paralel çizgilerle oluşturulmuş Yandaki bölgelerin alanları bir aritmetik dizi oluşturur.
= S, A(A ¿ BC)
A(BDEC) = 3S,
A(FGED) = 5S,
A(HKGF) = 7S,
Örnek
...
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3. Ölçme Örnek ABF ve DBC birer üçgen, |DB| |DB| = 6 cm, |BF| |BF| = 10 cm, |FC| = 5 cm, |AE| = 9 cm ve |EF| = 3 cm olduğuna göre |AD| = x kaç santimetredir? Bulalım.
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Konu Değerlendirme Soruları-2 1.
4.
ABC ve ADE birer dik üçgen |AD| = 4 cm |AE| = 6 cm |CD| = 8 cm |EB| = x
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Konu Değerlendirme Soruları-2 7.
ABC ve DEF birer üçgen,
D, E, K ve L bulundukları kenarların orta noktaları,
10.
d1 // d2 // d3
|AB| = 5 cm |EF| = 4 cm |ED| = 8 cm |BC| = x
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Konu Değerlendirme Soruları-2 13.
ABC ve DEC birer dik üçgen [DE] ⊥ [AC] [AB] ⊥ [DC] |AE| = 4 cm |EC| = 5 cm |BC| = 3 cm
16.
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Ünite Değerlendirme Tes 1.
ABD ve BCD birer birer dik üç üçgen gen [AB] ⊥ [BD] [BC] ⊥ [CD] |AB| = 3 cm |BC| = ò 15 cm
4.
ABC bir üç üçgen gen [AD] ⊥ [BC] |AB| = 10 cm |AC| = 17 cm |AD| = 8 cm
Yuka karırıda daki ki veri verile lere re göre |BC| kaç santimet-
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4. Trigonometri ve Uygulamaları Bunları Bilelim n sıfırdan farklı bir tam sayı a, b, c, d birer gerçek sayı olmak üzere, y = c + d . sinn(ax + b) y = c + d . cosn(ax + b) y = c + d . tann(ax + b) şeklindeki fonksiyonların periyodu r dır. Sadece n nin tek sayı olduğu durumlarda sinüs ve koa
sinüs fonksiyonunun periyodu 2r olur. a
Ô
Bir ifade birden fazla trigonometrik fonksiyon içeriyorsa, bu trigonometrik fonksiyonların esas periyotlarının en küçük ortak katının (EKOK ) tam katları bu fonksiyonun bir periyodudur.
Örnek 3
5
f (x) = sin (6x − 2 ) ve g (x) = tan (2 x + 6 ) fonksiyonlarının esas periyodunu bulalım.
Çözüm 3
f (x) = sin (6x − 2 ) & T =
2r a
5
g (x) = tan (2x + 6 ) & T =
T=
&
r
a
&
T=
2r 6 r
2
=
r
3
bulunur.
Örnek f(x) = cos2(6x+7) – tan3(2x + 4) fonksiyonunun esas periyodunu bulalım.
Çözüm 2
cos (6x + 7) & T1 = 3
tan (2x + 4)
&
T2 =
r
a
&
r
a
&
T1 = T2 =
r
6 r
2
olur.
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4. Trigonometri ve Uygulamaları Örnek g(x) = sin
+ cos(7 − x) − tan(3 − 2x)
fonksiyonunun esas periyodunu bulalım.
Çözüm sin
a 2x 1k +
cos (7 − x)
&
&
T1 =
2r a
&
T1 =
T2 =
2r a
&
T2 =
tan (3 − 2x) & T3 =
r
a
&
T3 =
2r = 4r 1 2 2r = 2r −1 r −
2
=
r
2
olur.
Bulunan periyotların EKOK 'u 'u alınır. EKOK
a
4r , 2r ,
r
2
k olduğundan esas periyot 4 dir. π
f(x) = a sin(bx + c) + k türündeki fonksiyonların grafikleri ve katsayılarının grafik üzerindeki etkilerini örneklerle inceleyelim.
Örnek Dinamik matematik/geometri yazılımını kullanarak y = sin(x + 3) + 2 ve y = sinx fonksiyonları nın grafiklerini çizelim ve inceleyelim. Dinamik matematik/geometri yazılımını açalım ve cebir sekmesini seçelim. Yazılımın Yazılımın giriş paneline gelelim. y = sin(x + 3) + 2 denklemini yazıp “Enter” tuşuna basalım. Tekrar Tekrar yazılımın giriş pa neline gelelim. y = sinx denklemini yazıp “Enter” tuşuna basalım.
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4. Trigonometri ve Uygulamaları Örnek Dinamik matematik/geometri yazılımını kullanarak y = 2.sinx – 3 ve y = sinx fonksiyonlarının gra fiklerini çizelim ve inceleyelim. Dinamik matematik/geometri yazılımını açalım ve cebir sekmesini seçelim. Yazılımın Yazılımın giriş paneli ne gelelim. y = 2 .sinx – 3 denklemini yazıp “Enter” tuşuna basalım. Tekrar Tekrar yazılımın giriş paneline gelelim. y = sinx denklemini yazıp “Enter” tuşuna t uşuna basalım.
Görüldüğü gibi y = 2.sinx – 3 fonksiyonunun grafiği, y = sinx fonksiyonunun grafiğinde y değerlerinin 2 katının alınıp y ekseni üzerinde 3 birim aşağı doğru ötelenmesiyle elde edilmiştir.
Örnek y = sin2x ve y = sinx fonksiyonlarının grafiklerini çizip inceleyelim.
Çözüm y = sin2x fonksiyonunun periyodu π dir dir.. y = sinx fonksiyonunun periyodu 2π olduğundan y = sinx in graği periyodu π olacak şekilde daraltılarak y = sin2x graği elde edilir. edilir. y = sin2x in graği π aralıklarla tekrarlanır. Buna göre grak aşağıdaki gibi olur.
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4. Trigonometri ve Uygulamaları Örnek Dinamik matematik/geometri yazılımını kullanarak y = cos(x + 1) + 2 ve y = cosx fonksiyonlarının grafiklerini çizelim ve inceleyelim. Dinamik matematik/geometri yazılımını açalım ve cebir sekmesini seçelim. Yazılımın Yazılımın giriş paneline gelelim. y = cos(x + 1) + 2 denklemini yazıp “Enter” tuşuna basalım. Tekrar yazılımın giriş paneline gelelim. y = cosx denklemini yazıp “Enter” tuşuna basalım.
Görüldüğü gibi y = cos(x + 1) + 2 fonksiyonunun grafiği, y = cosx fonksiyonunun grafiğinin x ekseni üzerinde 1 birim sola ve y ekseni üzerinde 2 birim yukarı ötelenmesiyle elde edilmiştir. edilmiştir.
Örnek y = cos2x ve y = cosx fonksiyonlarının grafiklerini çizip inceleyelim.
Çözüm y = cos2x fonksiyonunun periyodu π dir. y = cosx fonksiyonunun periyodu 2π olduğundan y = cosx in graği, periyodu π olacak şekilde daraltılarak y = cos2x in graği elde edilir. edilir. y = cos2x in graği π aralıklarla tekrarlanır. Buna göre grak aşağıdaki gibi olur.
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4. Trigonometri ve Uygulamaları Örnek Dinamik matematik/geometri matematik/geometri yazılımını kullanarak y = 3cosx – 1 ve y = cosx fonksiyonlarının grafiklerini çizelim ve inceleyelim. Dinamik matematik/geometri yazılımını açalım ve cebir sekmesini seçelim. Yazılımın Yazılımın giriş paneli ne gelelim. y = 3cosx – 1 denklemini yazıp “Enter” tuşuna basalım. Tekrar yazılımın giriş paneline gelelim. y = cosx denklemini yazıp “Enter” tuşuna tu şuna basalım.
Görüldüğü gibi y = 3cosx – 1 fonksiyonunun grafiği, y = cosx fonksiyonunun grafiğinde y değerlerinin 3 katının alınıp y ekseni üzerinde 1 birim aşağı doğru ötelenmesiyle elde edilmiştir.
Örnek Dinamik matematik / geometri yazılımını kullanarak y = 2 .tanx ve y = tanx fonksiyonlarının grafiklerini çizelim ve inceleyelim.
Çözüm Dinamik matematik / geometri yazılımını açalım ve cebir sekmesini tıklayalım. “Giriş” bölümüne
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4. Trigonometri ve Uygulamaları
y = 2tanx fonksiyonunun grafiği çizilirken y = tanx fonksiyonunun grafiğinde gra fiğinde y değerlerinin 2 katı alınarak elde edilmiştir.. edilmiştir..
Örnek Dinamik matematik / geometri yazılımını kullanarak y = tanx + 1 ve y = tanx fonksiyonlarının grafiklerini çizelim ve inceleyelim. Dinamik matematik / geometri yazılımını açalım ve cebir sekmesini tıklayalım. “Giriş” bölümüne gelelim. y = tanx + 1 yazarak “Enter” tuşuna basalım. Tekrar Tekrar yazılımın giriş paneline gelelim. y = tanx denklemini yazıp “Enter” tuşuna basalım.
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Konu Değerlendirme Soruları-2 1.
A = 5 – 2sinx olduğuna göre A nın alabileceği en büyük tam sayı değeri kaçtır? A) 3
2.
B) 4
B) –, +, +
B) cos α < 0
C) +, –, +
D) –, –, –
E) –, –, +
C) tan α > 0
D) tan α > sin α
E) cos θ > tan α
a = sin122o, b = sin133o, c = sin144o olduğuna göre a, b ve c nin doğru sıralaması aşağıdakilerden hangisidir? A) a < b < c
5.
E) 7
180o < α < θ < 270o olduğuna göre aşağıdakilerden hangisi yanlıştır? A) sin θ < 0
4.
D) 6
x = sin181 o, y = cos271o, z = tan271o olduğuna göre x, y ve z nin işaretlerinin doğru sıralaması aşağıdakilerden hangisidir? A) –, +, –
3.
C) 5
a
+
i
=
r 2
B) c < b < a
C) c < a < b
D) b < c <
E) a < c < b
olduğuna göre cos (4 (4α+5θ) + cos (4α+3θ) ifadesinin eşiti aşağıdakilerden hangisi-
dir? A) –2cosθ
B) 2cos θ
C) 0
D) 2
E) 1 + cos θ
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4. Trigonometri ve Uygulamaları
4.3. Trigonometrik Fonksiyonların Gerçek (Gerçekçi) Hayat Durumlarını Modellemede ve Problem Çözmede Kullanılması Trigonometrik fonksiyonları fonksiyonları gerçek/gerçekçi gerçek/gerçekçi hayat durumlarını modellemede ve problem çözmede de kullanırız. Trigonometrik fonksiyonların periyodikliğini içeren problemleri/örnekleri (ses dalgaları, kan basıncı, günlük sıcaklık değişimi, sinüzoidal alternatif akım vb.) inceleyelim.
Örnek Müzikal sesler y = a.sin(bx – c) denklemiyle ifade edilir. a değeri sesin genliğidir. a arttıkça sesin tonu yükselir. b değeri, 2π sayısı ile frekansın çarpımıdır. çarpımıdır. c değeri ise sesin titreşimindeki değişim miktarı olup faz farkı olarak adlandırılır. Mesela belirli bir sesin frekansı 15 Hz ve genliği 3 olduğunda, genlik 3 olduğundan a = 3 ve b = 2π.15 = 30π dir. Faz farkı olmadığından c = 0 dır. Bu durumda ses titreşiminin denklemi y = 3.sin(30πx) olur. Dinamik matematik / geometri yazılımını kullanarak ses titreşimi denkleminin grafiğini çizelim.
Çözüm Dinamik matematik / geometri yazılımını açalım ve “cebir” sekmesini tıklayalım. “Giriş” bölümüne gelelim. Bu kısma y = 3.sin(30 πx) denklemini yazalım ve “Enter” tuşuna basalım. Görüldüğü gibi sinüs eğrisiyle müzikal sesler modellenebil mektedir.
Örnek Düzgün dairesel hareket yapan bir cismin, bir tam dönüş yapması için gereken süreye periyot denir ve T ile gösterilir. Cismin birim zamanda yaptığı devir sayısına frekans denir ve f ile
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4. Trigonometri ve Uygulamaları Örnek
Müziğin Matematiği 19.yüzyıl matematikçilerinden Joseph Fourier müzikal seslerin basit periyodik sinüs fonksiyonlarıyla ifade edilebileceğini kanıtladı. Seslerin matematiksel olarak ifade edilmesi müziğin elektronik-bilgisayar ortamına aktarılmasında büyük rol oynamıştır. Mesela, gitar ve bağlama gibi enstrümanların çıkardığı çıkard ığı sesleri grafiklerle gösterebiliriz. Bir bağlama telinin titreşim grafiği sinüzoidal (sinüs veya kosinüs) grafiğine benzeyen bir eğri oluşturur oluşturur..
A
–A
Bir bağlama telini şekildeki gibi A uzunluğu kadar gerip bırakalım. A uzunluğunun zamana bağlı olarak değişimi sinüzoidal bir grafik oluşturur. Telin Telin bir titreşimini, yani gidiş ve gelişini 1/330 saniyede yaptığını kabul edersek grafiğin esas periyodu 1/330 olur. Bir saniyede 330 titreşim yapacağı için telin frekansı 330 Hz dir. Telin gerginliği ve uzunluğu değiştikçe periyot değişir. Bu şekilde frekans değiştirilerek farklı sesler elde edilebilir edilebilir..
Örnek Günlük sıcaklık değişimi kosinüs eğrisiyle modellenir. Eğrinin denklemi y = A.cos[a(x+b)] + B
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4. Trigonometri ve Uygulamaları Dinamik matematik / geometri yazılımını kullanarak sıcaklık değişiminin grafiğini çizelim. Dinamik matematik / geometri yazılımını açalım ve “cebir” sekmesini tıklayalım. “Giriş” bölümüne gelelim ve bu kısma y=1 y=10°. 0°.cos[ cos[ (x+ (x+9)] 9)]+20 +20 denklemini yazıp “Enter” tuşuna basalım. Görül -
12
düğü gibi kosinüs eğrisiyle günlük sıcaklık değişimi modellenebilmektedir. modellenebilmektedir.
Örnek Kalp, damar ve solunum yolu hastalıklarının teşhis ve tedavisinde kan basıncındaki değişimin ölçülmesi büyük bir ihtiyaçtır. Bu ölçümlerin yapılabilmesi için kullanılan elektronik araçların arka planında trigonometri ve periyodik fonksiyonlar kullanılır. Dinamik matematik / geometri yazılımını kullanarak bu durumu inceleyelim.
Çözüm
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4. Trigonometri ve Uygulamaları Örnek Bir öğrenci bir rezidansın yüksekliğini hesaplam sap lamak ak ist istem emekt ektedi edirr. Boy Boyuu 20 m o l a n bir ağacın gölgesinin uzunluğunun 15 m olduğu anda rezidansın gölgesi 45 m olduğuna göre öğrenci rezidansın boyunu kaç metre hesaplamıştır? Bulalım.
Çözüm Bu problemi aşağıdaki gibi ABC ve DEF dik üçgenleriyle modelleyelim. Güneş ışınları yer düzlemiyle her D noktada aynı açıyı yaptığı için = m(D ëEF) = x dir. m(Aë BC) A
h = 60 m 20 m x B
AC
⇒ ABC de
tanx =
= BC
20 15
=
4
olur.
3
x 15 m
C
E
45 m
F
DF
⇒ ABC de
tanx =
= EF
4 &
3
h =
45
&
h
=
h 45
60 metre bulunur.
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4. Trigonometri ve Uygulamaları Örnek
Yandaki resimde bir elma ağacının karşısında duran Ahmet'in göz hizasının elma ağacına olan uzaklığı 3 m ve göz hizasıyla ağacın kökünü ve tepesini görebildiği açı da sırasıyla 30° ve 45° dir d ir.. Buna göre ağacın boyu kaç metredir? Bulalım.
Çözüm Bu problemi yandaki ABC ve DBC dik üçgenleri ile modelleyelim. ABC dik üçgeninde tanjant fonksiyonunun değerinden,
tan45°=
AB ⇒
BC
1=
AB ⇒
AB = 3 m olur.
3
DBC dik üçgeninde tanjant fonksiyonunun değerinden,
tan30°=
BD
1 ⇒
BC
3
=
BD ⇒
BD =
3
Bu değerlerden ağacın boyu, |AD| = h = |AB| + |BD| = 3 + ñ 3 metre bulunur.
3 m ol olu ur.
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Ünite Değerlendirme Tes ë ) =14° m( A =14° 24 ı
1.
5.
A
) =12° 17 ı m( ëB
x
ë ) farkı aşağıolduğuna göre 2.m( ë B ) – m( A dakilerden hangisidir? A) 10° 10 ı
B) 10° 12 ı
ı 12° 12
ı
D)
C) 10° 14 ı
π
E) 12° 10
40o
3
B
C
Yukarıda Y ukarıda verilen ABC üçgeninde
^h
^h
m A = x, m B = W
W
r
^h
ve ve m C = 40 o olW
3
duğuna göre x kaç derecedir? 2.
A) 60
1071 dakikalık açının derece ve dakika türünden eşiti aşağıdakilerden hangisidir? A) 17° 41 41 ı
B) 17° 48 ı
D) 17° 58 ı
E) 18° 51 ı
3
B
II
III
D) 90
E) 100
330 derecelik açı kaç radyandır? A) 11r
I
C) 80
C) 17° 51 ı
6.
3.
B) 70
D) B
11r 6
B) 10r 3
E)
3r 3
C) 5r 3
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Ünite Değerlendirme Tes 8.
− 47r radyanın esas ölçüsü aşağıdakilerden
2 o 2 o 2 o 2 o 13. co cos s 10 + co cos s 20 + co cos s 30 + ... + co cos s 80
4
hangisidir? A)
r 2
B)
işleminin sonucu kaçtır? r 3
C) r 4
D)
r
5
E)
r
A) 2
6
B) 3
14. si sinx nx − co cosx sx = 9.
− 1400o lik açının esas ölçüsü kaç derecedir?
A) 40
B) 60
C) 80
D) 100
ifadesinin alabileceği en büyük değer kaçtır? C) 7
D) 8
2
olduğu
^
E) 6
sinx + cosx
h ifa 2
-
E) 120
10. 3 $ si sinx − 2 $ co cosy + 4
B) 6
1
D) 5
desinin eşiti kaçtır? A)
A) 5
C) 4
E) 9
1 4
B)
C) 3
1
4
2
15.
sin(-210o) cos(-78o) tan(-110o) sin(-192o)
D)
3
E)
2
I
II
III
IV
V
+ + + +
+ + +
+ +
+
-
-
-
-
-
-
-
-
-
-
7 4
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Ünite Değerlendirme Tes 5r 17. x = tan 4
y = cos 5r 6
oduğuna göre x, y, z nin işaretlerinin doğru sıralaması aşağıdakilerden hangisidir? A) +, -, -
B) +, -, +
D) +, +, +
E) -, +, -
C) -, +, +
Bir ABC üçgeninde,
^
h
^h
^
sin A + B – sin C + cos A + B + C W
W
19.
r
2
1
B) 0
cos ë B > 0, tan ëB < 0
W
W
W
C) 1
D) 2
ë sin ve tan ë C nin işaolduğuna göre cos A, sin ëB retleri sırasıyla aşağıdakilerden hangisidir? A) –, –, +
B) –, –, –
D) –, +, +
E) +, +, +
^h
22. f x = sin 3
h
W
ifadesinin eşiti aşağıdakilerden hangisidir? A) -1
sin A ë < 0, tan A ë > 0
< 0, cos ë C > 0 sin ëC
z = sin 5r 3
18.
21.
E) 3
x1y1r
olduğuna göre aşağıdaki ifadelerden han
c
x−5 2
m
+ cos
2
C) +, –, +
a
x+e 4
k
fonksiyonunun periyodu aşağıdakilerden hangisidir? A)
r 2
B)
r
C)
2r
D)
r
E) 8r
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KONU VE ÜNİTE DEĞERLENDİRME SORULARININ CEVAP ANAHTARI
1. ÜNİTE: GRAFİKLERİN VE TABLOLARIN YORUMLANMASI Ünite Değerlendirme 1. D
2. C
3. C
4. D
5. E
6. B
11. B
12. D
13. E
14. E
15. B
16. E
7. E
8. A
9. B
2. ÜNİTE: ÜSTEL FONKSİYONLAR VE UYGULAMALARI Konu Değerlendirme-1
10. A
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4. ÜNİTE: TRİGONOMETRİ VE UYGULAMALARI Konu Değerlendirme-1 1. A
2. C
3. D
4. B
5. B
6. A
4. B
5. B
6. E
7. D
Konu Değerlendirme-2 1. E
2. A
3. E
Ünite Değerlendirme-2 1. A
2. C
3. B
4. D
5. C
6. D
7. D
8. C
9. A
10. E
11. A
12. B
13. C
14. E
15. A
16. A
17. A
18. A
19. C
20. D
21. B
22. D
23. D
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SÖZLÜK -Aaçı açıortay ağırlık merkezi alan analitik düzlem apsis apsisler ekseni
: Başlangıç noktaları ortak olan iki ışının birleşimi. : Bir açıyı iki eş ölçülü açıya ayıran ışın. : Bir üçgenin kenarortayların kenarortaylarının ın kesim noktası. : Bir yüzeyin bölümü. : Üzerine koordina koordinatt sistemi yerleştirilmiş düzlem. : Koordinat düzlemindeki bir noktanın birinci bileşeni (x). : Koordinat düzlemini oluşturan yatay eksen (x ekseni).
-Bbenzerlik
: İki üçgende köşelerin eşlenmesine göre karşılıklı açılarının eş ve karşılıklı
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dik kenar diklik merkezi dik üçgen
: Bir dik üçgende, dik açıyı oluşturan kenarlardan her biri. : Üçgende yüksekliklerin kesim noktası. : Herhangi bir açısının ölçüsü 90 derece olan üçgen.
-Eeğim eğim açısı eksen eş üçgenler eşkenar üçgen
Düzlemde bir doğrunun x ekseni ile pozitif yönde yaptığı açının tanjantı. : Düzlemde bir doğrunun x ekseni ile pozitif yönde yaptığı açı. : Koordinat düzlemlerini oluşturan sayı doğrularınd doğrularından an her biri. : Karşılıklı tüm elemanlarının ölçüleri eşit olan üçgenler. : Kenar uzunluğu eş olan üçgen.
:
-Ffonksiyon fonksiyonun değer kümesi
: Bir kümenin her elemanını, bir başka kümenin elemanları ile eşleyen ve elemanı birden fazla elemana eşlemeyen bağıntı. : f : A
B fonksiyonundaki B kümesi.
→
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kenarortay koordinat sistemi
: Bir üçgendek üçgendekii bir kenarın orta noktasını karşı bölgeye birleştiren doğru parçası. : İki sayı doğrusunu doğrusunun n sıfır noktaları çakışacak ve birbirine dik olacak şekilde konumlandırılması konumlan dırılması ile oluşan sistem.
-Mmaksimum değer minimum değer
: Bir ifadenin aldığı en büyük değer. : Bir ifadenin aldığı en küçük değer.
-Oordinat orijin orta dikme orta nokta orta taban ortanca
: Koordinat düzlemindeki bir noktanın ikinci bileşeni (y). : Koordinat eksenleri eksenlerinin nin başlangıç noktası. : Bir doğru parçasının orta noktasına dik olan doğru. : Bir doğru parçasının uç noktaları noktalarına na eşit uzaklıkta bulunan nokta. : Bir üçgenin iki kenarının orta noktalarını birleştiren doğru parçası. : Küçükten büyüğe sıralanmış veri sonuçlarının tam ortasında yer alan değer.
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KAYNAKÇA Brannan D.A, Esplen M.F., Gray J.J., (2012), Geometry, Geometry, Second Second Edition, Cambridge University Press. Larson R., (2010), Algebra, Algebra, Fifth Fifth Edition, Brooks/Cole Cengage Learning, USA. Larson R., (2013), Trigonometry, Trigonometry, Eighth Eighth Edition, Wadsworth Publishing. Publishing.
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SEMBOL VE GÖSTERİMLER ∧
ve
br
∨
veya
cm
⇒
m
⇔
km (ancak ve ancak)
dk
metre