Question

the given unit circle has been divided into eight equal arcs, corresponding to t - values of 0, $\frac{pi}{4}$, $\frac{pi}{2}$, $\frac{3pi}{4}$, $pi$, $\frac{5pi}{4}$, $\frac{3pi}{2}$, $\frac{7pi}{4}$, and $2pi$. complete parts (a) and (b).
(a) $sin\frac{pi}{4}=$ $\frac{sqrt{2}}{2}$ (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)
a. the solution is $\frac{sqrt{2}}{2}$
b. the solution is undefined.
(b) use periodic properties and the answer from part (a) to find the value of $sin\frac{65pi}{4}$. select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. $sin\frac{65pi}{4}=$ (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)
b. the solution is undefined.

Explanation:

Step1: Recall the period of sine function

The period of $y = \sin(x)$ is $2\pi$. We can write $\frac{65\pi}{4}$ as $\frac{64\pi+\pi}{4}=16\pi+\frac{\pi}{4}$. Since $16\pi$ is a multiple of the period $2\pi$ of the sine - function, $\sin(\frac{65\pi}{4})=\sin(16\pi + \frac{\pi}{4})$. According to the periodic property $\sin(x + 2k\pi)=\sin(x)$ where $k\in\mathbb{Z}$, and here $k = 8$ (because $16\pi=2\pi\times8$), so $\sin(\frac{65\pi}{4})=\sin(\frac{\pi}{4})$.

Step2: Find the value of $\sin(\frac{\pi}{4})$

From the unit - circle, for the angle $\theta=\frac{\pi}{4}$, the coordinates of the corresponding point on the unit - circle are $(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2})$. The sine of an angle in the unit - circle is the $y$ - coordinate of the corresponding point. So $\sin(\frac{\pi}{4})=\frac{\sqrt{2}}{2}$.

Answer:

A. $\sin\frac{65\pi}{4}=\frac{\sqrt{2}}{2}$