Question

Kuis ⁽¹⁶⁴⁾
Berapa hasil dari

[tex]\huge\boxed{\rm\frac{~~~~\frac{sin\left(\frac{\pi}{180}\right)sin\left(\frac{\pi}{90}\right)~...~sin\left(\frac{89\pi}{180}\right)}{cos\left(\frac{\pi}{180}\right)cos\left(\frac{\pi}{90}\right)~...~cos\left(\frac{89\pi}{180}\right)}~~~~}{\frac{tan\left(\frac{\pi }{180}\right)tan\left(\frac{\pi}{90}\right)~...~tan\left(\frac{89\pi}{180}\right)}{cot\left(\frac{\pi}{180}\right)cot\left(\frac{\pi}{90}\right)~...~cot\left(\frac{89\pi}{180}\right)}}}[/tex]

Answer 1

Penjelasan dengan langkah-langkah:

Trigonometri

Identitas trigonometri

[tex]\boxed{\rm\frac{\sin\alpha}{\cos\alpha}=\tan\alpha}[/tex]

[tex]\begin{array}{c|c}\rm\frac{\pi}{180}&\rm1^o \\ \\\rm\frac{\pi}{90}&\rm2^o\\ \\\vdots&\vdots\\ \\ \frac{89\pi}{180}&\rm89^o\end{array}[/tex]

Sehingga diperoleh

[tex]\left(\large\rm\frac{\tan(1^o)\tan(2^o)\cdots\tan(89^o)}{\frac{\tan(1^o)\tan(2^o)\cdots\tan(89^o)}{\cot(1^o)\cot(2^o)\cdots\cot(89^o)}}\right)[/tex]

Maka

[tex]\begin{aligned}&=\cancel{\rm\tan(1^o)\tan(2^o)\cdots\tan(89^o)}\times\frac{\cot(1^o)\cot(2^o)\cdots\cot(89^o)}{\cancel{\tan(1^o)\tan(2^o)\cdots\tan(89^o)}}\\&=\rm \cot(1^o)\cot(2^o)\cdots\cot(89^o)\\&=\rm \cot(1^o).\cot(89^o)\times\cot(2^o)\cot(88^o)\cdots\cot(45°)\\&=\rm 1\times1\times\cdots\times1\\&=\rm 1\end{aligned}[/tex]

[tex]\therefore[/tex] Maka hasil dari operasi Trigonometri diatas adalah 1