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Chapter 4: Problem 85

In Exercises \(61-86,\) use reference angles to find the exact value of each expression. Do not use a calculator. $$\sin \left(-\frac{17 \pi}{3}\right)$$

Short Answer

Expert verified
\(\sin \left(-\frac{17 \pi}{3}\right) = \frac{\sqrt{3}}{2}\)

Step by step solution

01

Normalize the Angle

Before finding the sin value of the angle we need to normalize the angle because the range of sine function is -\(\pi\) to \(\pi\). We can do this using the formula \(Normalized Angle = Initial Angle modulo 2\pi\). Now normalize the angle by subtracting or adding multiples of \(2\pi\) from \(-\frac{17\pi}{3}\) until it lies in the range of [-\(\pi\), \(\pi\)]. By doing so, we get the normalized angle to be \(\frac{\pi}{3}\).
02

Find the Reference Angle

Calculate the reference angle. The reference angle is always be a positive acute angle, and it is found by taking the absolute value of the angle. Since our normalized angle \(\frac{\pi}{3}\) is already positive and acute, the reference angle is also \(\frac{\pi}{3}\). Reference angle = |Normalized Angle| = \(\frac{\pi}{3}\).
03

Find the sine of the Reference Angle

Now, that we have our reference angle, we can find the sine of the reference angle. We know the exact values of sin function for popular angles like \(0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}\). Hence, \(\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}\).

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