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Rabu, 24 Juli 2013

Trigonometri #6

6.       Batasan nilai perbandingan trigonometri
Batas-batas nilai perbandingan trigonometri terdefinisi sebagai berikut:

latex {\color{Blue} \begin{array}{rcl}-1\leqslant sinx\leqslant 1\qquad &\Rightarrow& \qquad cscx \geq 1\ atau\ cscx< -1\\\\-1\leqslant cosx\leqslant 1\qquad &\Rightarrow& \qquad secx \geq 1\ atau\ secx< -1\\\\-\infty \leqslant tanx\leqslant +\infty \qquad &\Rightarrow& \qquad -\infty\leq cotx \leq +\infty\end{array}}


Contoh-contoh soal

1.        Diketahi latex sinx=\frac{p-2}{8-p}. Tentukan batas-batas nilai latex p.
Jawaban:
latex sinx=\frac{p-2}{8-p}\qquad \Rightarrow \qquad -1\leq \frac{p-2}{8-p}\leq 1

Bagian pertama:
latex \large{\begin{array}{rcl}\frac{p-2}{8-p}&\leq &1 \\\\\frac{p-2}{8-p}-1&\leq &0\\\\\frac{p-2-(8-p)}{8-p}&\leq &0\\\\\frac{2p-10}{8-p}&\leq &0\\\\\frac{2(p-5)}{8-p}&\leq &0\\\\p\leq 5&atau&p> 8 \end{array}}

Bagian kedua:
latex \large{\begin{array}{rcl}\frac{p-2}{8-p}&\geq &-1 \\\\\frac{p-2}{8-p}+1&\geq &0\\\\\frac{p-2}{p-8}-1&\leq &0\\\\\frac{p-2-(p-8)}{p-8}&\leq &0\\\\\frac{6}{p-8}&\leq &0\\\\p&\leq &8 \end{array}}

Karena latex p> 8 dan latex p< 8 saling menggugurkan maka batasan nilai latex p yang memenuhi adalah latex p\leq 5.

2.      Untuk sudut latex \varphi dalam kuadran III diketahui latex tan\varphi =\frac{1}{2}\sqrt{2}. Carilah perbandingan trigonometri lainnya dari latex \varphi.
Jawaban:
(1)     latex tan\varphi =\frac{1}{2}\sqrt{2}
(2)   latex cot\varphi =\frac{2}{\sqrt{2}}=\sqrt{2}
(3)   latex sec^2\varphi =1+tan^2\varphi =1+\frac{1}{4}.2=\frac{6}{4}
latex sec\varphi =-\sqrt{\frac{6}{4}}=-\frac{1}{2}\sqrt{6}
(4)   latex cos\varphi =-\frac{2}{\sqrt{6}}=-\frac{1}{3}\sqrt{6}
(5)   latex cos^2\varphi +sin^2\varphi =1
latex \frac{6}{9}+sin^2\varphi =1
latex sin^2\varphi =1-\frac{6}{9}=\frac{3}{9}
latex sin\varphi =-\sqrt{\frac{3}{9}}=-\frac{1}{3}\sqrt{3}
(6)   latex csc\varphi =-\frac{3}{\sqrt{3}}=-\sqrt{3}

3.      Untuk sudut latex \delta dalam kuadran IV, nyatakan semua perbandingan trigonometri dalam latex cot\delta .
Jawaban:
(1)     latex cot\delta = cot\delta
(2)   latex tan\delta = \frac{1}{cot\delta }
(3)   latex csc^2\delta = cot^2\delta +1
latex csc\delta = -\sqrt{cot^2\delta +1}
(4)   latex sin\delta = -\frac{1}{\sqrt{cot^2\delta +1}}= -\frac{\sqrt{cot^2\delta +1}}{cot^2\delta +1}
(5)   latex cos^2\delta +sin^2\delta =1
latex cos^2\delta +\frac{1}{cot^2\delta +1}=1
latex cos^2\delta +\frac{1}{cot^2\delta +1}=1
latex cos\delta =-\frac{cot\delta }{\sqrt{cot^2\delta +1}}=-\frac{cot\delta \sqrt{cot^2\delta +1}}{cot^2\delta +1}
(6)     latex sec\delta =-\frac{\sqrt{cot^2\delta +1}}{cot\delta }

Adjie Gumarang Pujakelana, 2013


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