the given unit circle has been divided into eight equal arcs, corresponding to t - values of 0…

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).\n(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.)\na. the solution is $\frac{sqrt{2}}{2}$\nb. the solution is undefined.\n(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.\na. $sin\frac{65pi}{4}=$ (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)\nb. the solution is undefined.

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).\n(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.)\na. the solution is $\frac{sqrt{2}}{2}$\nb. the solution is undefined.\n(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.\na. $sin\frac{65pi}{4}=$ (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)\nb. the solution is undefined.

Answer

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