Question

find the exact values of the six trigonometric functions of the given angle. if any are not defined, say not defined. do not use a calculator.
\frac{23\pi}{2}
what is the value of \sin\frac{23\pi}{2}? select the correct choice and, if necessary, fill in the answer box to complete your choice.
a. \sin\frac{23\pi}{2}=\square
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)
b. the function is not defined

Explanation:

Step1: Simplify the angle

We know that the sine function has a period of \(2\pi\). So, we can find an equivalent angle within one period (\(0\) to \(2\pi\)) by subtracting multiples of \(2\pi\).

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Since \(\sin(x + 2k\pi)=\sin x\) for any integer \(k\), and here \(k = 6\) (because \(12\pi=2\pi\times6\)), we have \(\sin\frac{23\pi}{2}=\sin(12\pi-\frac{\pi}{2})\)

Step2: Use the sine subtraction formula

The formula \(\sin(A - B)=\sin A\cos B-\cos A\sin B\). Here \(A = 12\pi\) and \(B=\frac{\pi}{2}\). We know that \(\sin(12\pi)=0\) and \(\cos(12\pi) = 1\)

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Answer:

\(\sin\frac{23\pi}{2}=-1\)