Question

find the exact values of the six trigonometric functions of the given angle. if any are not defined, say
ot defined.\ do not use a calculator. what is the value of sin(7π/2)? select the correct choice and, if necessary, fill in the answer box to complete your choice. a. sin(7π/2)= - 1 (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.) b. the function is not defined. what is the value of cos(7π/2)? select the correct choice and, if necessary, fill in the answer box to complete your choice. a. cos(7π/2)= (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.) b. the function is not defined.

Explanation:

Step1: Rewrite the angle

First, rewrite $\frac{7\pi}{2}$ as $\frac{7\pi}{2}= 3\pi+\frac{\pi}{2}$.

Step2: Analyze the sine - function property

We know that $\sin(x + 2k\pi)=\sin x$ for any real - number $x$ and integer $k$, and $\sin(x+\pi)=-\sin x$.
$\sin(\frac{7\pi}{2})=\sin(3\pi+\frac{\pi}{2})$. Since $\sin(A + B)=\sin A\cos B+\cos A\sin B$, when $A = 3\pi$ and $B=\frac{\pi}{2}$, $\sin(3\pi+\frac{\pi}{2})=\sin3\pi\cos\frac{\pi}{2}+\cos3\pi\sin\frac{\pi}{2}$.
We know that $\sin3\pi = 0$, $\cos3\pi=-1$, $\cos\frac{\pi}{2}=0$, and $\sin\frac{\pi}{2}=1$.
So $\sin(3\pi+\frac{\pi}{2})=- 1$.

Step3: Analyze the cosine - function property

For $\cos(\frac{7\pi}{2})=\cos(3\pi+\frac{\pi}{2})$. Using the formula $\cos(A + B)=\cos A\cos B-\sin A\sin B$, when $A = 3\pi$ and $B=\frac{\pi}{2}$, $\cos(3\pi+\frac{\pi}{2})=\cos3\pi\cos\frac{\pi}{2}-\sin3\pi\sin\frac{\pi}{2}$.
Since $\sin3\pi = 0$ and $\cos3\pi=-1$, $\cos\frac{\pi}{2}=0$, $\sin\frac{\pi}{2}=1$, we have $\cos(3\pi+\frac{\pi}{2}) = 0$.

Answer:

For $\sin\frac{7\pi}{2}$, the answer is A. - 1.
For $\cos\frac{7\pi}{2}$, the answer is B. The function is not defined.