Latihan Matematika Wajib Kelas X Fungsi Trigonometri dan Bilangan Real
# 3
Pilgan

Nilai dari ekspresi sin⁡θ+cos⁡θ\sin\theta+\cos\theta jika θ=−16π\theta=-\frac{1}{6}\pi adalah ....

A

122− 12\frac{1}{2}\sqrt{2}-\ \frac{1}{2}

B

−123− 12-\frac{1}{2}\sqrt{3}-\ \frac{1}{2}

C

123+ 12\frac{1}{2}\sqrt{3}+\ \frac{1}{2}

D

123− 12\frac{1}{2}\sqrt{3}-\ \frac{1}{2}

E

12− 123\frac{1}{2}-\ \frac{1}{2}\sqrt{3}

Pembahasan:

Perlu diingat identitas trigonometri pada sudut-sudut berlawanan tanda, yaitu

sin⁡(−θ)=−sin⁡θ\sin\left(-\theta\right)=-\sin\theta

cos⁡(−θ)=cos⁡θ\cos\left(-\theta\right)=\cos\theta

tan⁡(−θ)=−tan⁡θ\tan\left(-\theta\right)=-\tan\theta

Pada soal diketahui sin⁡θ+cos⁡θ\sin\theta+\cos\theta dengan θ=−16π\theta=-\frac{1}{6}\pi. Diperoleh

sin⁡θ+cos⁡θ=sin⁡(−16π)+cos⁡(−16π)\sin\theta+\cos\theta=\sin\left(-\frac{1}{6}\pi\right)+\cos\left(-\frac{1}{6}\pi\right)

⇔sin⁡θ+cos⁡θ=−sin⁡16π+cos⁡16π\Leftrightarrow\sin\theta+\cos\theta=-\sin\frac{1}{6}\pi+\cos\frac{1}{6}\pi

⇔sin⁡θ+cos⁡θ=−sin⁡(16π180°π)+cos⁡(16π180°π)\Leftrightarrow\sin\theta+\cos\theta=-\sin\left(\frac{1}{6}\pi\frac{180\degree}{\pi}\right)+\cos\left(\frac{1}{6}\pi\frac{180\degree}{\pi}\right)

⇔sin⁡θ+cos⁡θ=−sin⁡(16180°)+cos⁡(16180°)\Leftrightarrow\sin\theta+\cos\theta=-\sin\left(\frac{1}{6}180\degree\right)+\cos\left(\frac{1}{6}180\degree\right)

⇔sin⁡θ+cos⁡θ=−sin⁡(30°)+cos⁡(30°)\Leftrightarrow\sin\theta+\cos\theta=-\sin\left(30\degree\right)+\cos\left(30\degree\right)

⇔sin⁡θ+cos⁡θ=−12+123\Leftrightarrow\sin\theta+\cos\theta=-\frac{1}{2}+\frac{1}{2}\sqrt{3}

⇔sin⁡θ+cos⁡θ=123−12\Leftrightarrow\sin\theta+\cos\theta=\frac{1}{2}\sqrt{3}-\frac{1}{2}

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