Problem 67
Question
In Exercises \(61-86,\) use reference angles to find the exact value of each expression. Do not use a calculator. $$\sin \frac{2 \pi}{3}$$
Step-by-Step Solution
Verified Answer
The exact value of \(\sin 2\pi/3\) is \(\sqrt{3}/2\).
1Step 1: Identifying the Quadrant
Given θ = \(2\pi/3\) which exists in the second quadrant where sine function is positive. Also, we know that θ = \(2\pi/3\) is equivalent to 120° in degree
2Step 2: Finding the Reference Angle
The reference angle r is found by subtracting 180° from θ if θ is in the second quadrant. Hence, reference angle r = 180° - 120° = 60°. In terms of radians, r = \(π - 2\pi/3 = \pi/3\)
3Step 3: Calculating the Exact Value
The exact value of sine of the reference angle is equal to the sine of θ because sine is positive in the second quadrant. Using the unit circle values; \(\sin\pi/3\) is calculated as \(\sqrt{3}/2\). Hence \(\sin 2\pi/3 = \sin\pi/3 = \sqrt{3}/2\).
Other exercises in this chapter
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