Question

assume angles are in degrees.
\\(\sin(354)\\) is the same as
(a) \\(\sin(6)\\)
(b) \\(\sin(-85)\\)
(c) \\(\sin(85)\\)
(d) dont know
(e) \\(\sin(-6)\\)

Explanation:

Step1: Use angle - periodicity property

The sine function has a period of 360 degrees, i.e., $\sin(x)=\sin(x + 360n)$ where $n\in\mathbb{Z}$. We can write $354=360 - 6$. So, $\sin(354)=\sin(360 - 6)$.

Step2: Apply the trigonometric identity

The identity $\sin(A - B)=\sin A\cos B-\cos A\sin B$. When $A = 360^{\circ}$ and $B = 6^{\circ}$, $\sin(360^{\circ}) = 0$ and $\cos(360^{\circ})=1$. Then $\sin(360 - 6)=\sin360\cos6-\cos360\sin6=0\times\cos6 - 1\times\sin6=-\sin6=\sin(- 6)$ (since $\sin(-\theta)=-\sin\theta$).

Answer:

E. $\sin(-6)$