Question

Q.
Tentukan nilai dari
[tex]\frac{Cos\:95°+Cos\:25°}{Sin\:95°+Sin\:25°}=...[/tex]​

Answer 1

Verified answer

[tex]\begin{aligned}&\frac{\cos95^\circ+\cos25^\circ}{\sin95^\circ+\sin25^\circ}=\boxed{\,\bf\frac{1}{3}\sqrt{3}\,}\end{aligned}[/tex]
 

Penjelasan dengan langkah-langkah:

Trigonometri

Kita akan menentukan nilai dari

[tex]\dfrac{\cos95^\circ+\cos25^\circ}{\sin95^\circ+\sin25^\circ}[/tex]

Perhatikan identitas trigonometri berikut ini.

[tex]\begin{aligned}&\bullet\ \cos\alpha+\cos\beta\\&\quad=2\cos\left(\frac{\alpha+\beta}{2}\right)\cos\left(\frac{\alpha-\beta}{2}\right)\\&\bullet\ \sin\alpha+\sin\beta\\&\quad=2\sin\left(\frac{\alpha+\beta}{2}\right)\cos\left(\frac{\alpha-\beta}{2}\right)\\\end{aligned}[/tex]

Maka:

[tex]\begin{aligned}&\frac{\cos\alpha+\cos\beta}{\sin\alpha+\sin\beta}\\&=\frac{\cancel{2}\cos\left(\dfrac{\alpha+\beta}{2}\right)\cancel{\cos\left(\dfrac{\alpha-\beta}{2}\right)}}{\cancel{2}\sin\left(\dfrac{\alpha+\beta}{2}\right)\cancel{\cos\left(\dfrac{\alpha-\beta}{2}\right)}}\\&=\frac{\cos\left(\dfrac{\alpha+\beta}{2}\right)}{\sin\left(\dfrac{\alpha+\beta}{2}\right)}\,=\,\cot\left(\frac{\alpha+\beta}{2}\right)\end{aligned}[/tex]

Oleh karena itu:

[tex]\begin{aligned}&\frac{\cos95^\circ+\cos25^\circ}{\sin95^\circ+\sin25^\circ}\\&=\cot\left(\frac{95^\circ+25^\circ}{2}\right)\\&=\cot\left(\frac{120^\circ}{2}\right)=\cot\left(60^\circ\right)\\&=\boxed{\,\bf\frac{1}{3}\sqrt{3}\,}\end{aligned}[/tex]

Atau, jika tidak ingat nilai cot(60°), bisa kita selesaikan dengan cos(60°) / sin(60°).
[tex]\blacksquare[/tex]