2. Assalamu'alaikum.... bantu kak mengerjakan soal matematika trigonometri... no 13,14,15,16,17,18,1.... Question from @riorocker10

October 2018 1 168 Report

2. Assalamu'alaikum.... bantu kak mengerjakan soal matematika trigonometri...
no 13,14,15,16,17,18,19,20,21,22

fatihahalhusna Nomor 13 (E)

$\displaystyle \sin x=\frac{1}{3}\sqrt{2}\rightarrow \cos x=\frac{1}{3}\sqrt{7}\\\\2\cos (\pi-x)+\sin(\frac{\pi}{2}+x)=-2\cos x+\cos x=-\cos x=\boxed{-\frac{1}{3}\sqrt{7}}$

Nomor 14 (???)
$\sin\theta+\cos\theta=\frac{1}{2}\\1+2\sin\theta\cos\theta=\frac{1}{4}\\\sin\theta\cos\theta=-\frac{3}{8}$
$$\begin {align} \sin^3\theta+\cos^3\theta&=(\sin\theta+\cos\theta)^3-3\sin\theta\cos\theta(\sin\theta+\cos\theta)\\&=(\frac{1}{2})^3-3(-\frac{3}{8})(\frac{1}{2})\\&=\boxed{\frac{11}{16}}
\end$

Nomor 15 (A)
$\tan\alpha=\sqrt{2}\rightarrow \cos\alpha=\frac{1}{3}\sqrt{3}\\\\\cos(\pi+\alpha)=-\cos\alpha=\boxed{-\frac{1}{3}\sqrt{3}}$

Nomor 16 (C)
$\displaystyle \frac{-\sin\theta.\sec\theta.(-\tan\theta)}{\sec\theta.(-\sin\theta)(-\tan\theta)}=\boxed{1}$

Nomor 17 (C)

$x^4=a^2\cot^2\alpha+2ab\cot\alpha\csc\alpha+b^2\csc^2\alpha\\y^4=b^2\cot^2\alpha+2ab\cot\alpha\csc\alpha+a^2\csc^2\alpha\\x^4-y^4=(a^2-b^2)\cot^2\alpha-(a^2-b^2)\csc^2\alpha\\~~~~~~~~~~~=(a^2-b^2)(\cot^2\alpha-\csc^2\alpha)\\~~~~~~~~~~~=\boxed{b^2-a^2}$

Nomor 18 (C)

$\displaystyle y=2+3\cos^2x\\y_{maks}=2+3=\boxed{5}$

Nomor 19 (A)
$\displaystyle y=2+3\sin^2x\\y_{min}=2+0=\boxed{2}$

Nomor 20 (D)
$y=1-\sin^2x\\y_{min}=1-1=\boxed{0}$

Nomor 21 (A)
$~~~\log\tan5.\tan6.\tan7...\tan85\\=\log\tan5\tan85\tan6\tan84...\tan44\tan46\\=\log1.1.1.1.1...\\=\log1\\=\boxed{0}$

Nomor 22 (C)
$\displaystyle \tan x=k\\\cos x=\frac{1}{\sqrt{k^2+1}}\\\\\sin x=\frac{k}{\sqrt{k^2+1}}\\\\\sin x\cos x=\boxed{\frac{k}{k^2+1}}$

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