Question

watch the video and then solve the problem given below. click here to watch the video. use the unit circle to find the value of sin 2π and periodic properties of trigonometric functions to find the value of sin 4π.

Explanation:

Step1: Recall sine - unit circle relationship

On the unit circle, for an angle $\theta$, $\sin\theta$ is the $y$ - coordinate of the point where the terminal side of the angle intersects the unit circle. For $\theta = 2\pi$, the terminal side of the angle lies on the positive $x$ - axis at the point $(1,0)$. So, $\sin(2\pi)=0$.

Step2: Use periodicity of sine function

The sine function $y = \sin(x)$ has a period of $2\pi$, which means $\sin(x + 2k\pi)=\sin(x)$ for all real $x$ and all integers $k$. Let $x = 2\pi$ and $k = 1$. Then $\sin(4\pi)=\sin(2\pi + 2\pi)$. Since the period of $\sin(x)$ is $2\pi$, $\sin(4\pi)=\sin(2\pi)$.

Answer:

$\sin(2\pi)=0$, $\sin(4\pi)=0$