12. Trigonometric Functions

Trigonometric Functions on the Unit Circle

12. Trigonometric Functions

Trigonometric Functions on the Unit Circle

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Multiple Choice

For each expression, identify which coterminal angle to use & determine the exact value of the expression.
$\sin\frac{7\pi}{3}$

$\frac12$

$2$

$\frac{\sqrt3}{2}$

$\frac{2\sqrt3}{3}$

Verified step by step guidance

Step 1: Understand the problem. The goal is to evaluate the trigonometric expression $ \sin\left(\frac{7\pi}{3}\right) $. Since the angle $ \frac{7\pi}{3} $ is greater than $ 2\pi $, we need to find a coterminal angle within the interval $ [0, 2\pi) $. Coterminal angles are found by adding or subtracting multiples of $ 2\pi $.

Step 2: Subtract $ 2\pi $ from $ \frac{7\pi}{3} $ to find the coterminal angle. Express $ 2\pi $ with a denominator of 3: $ 2\pi = \frac{6\pi}{3} $. Subtract: $ \frac{7\pi}{3} - \frac{6\pi}{3} = \frac{\pi}{3} $. Thus, the coterminal angle is $ \frac{\pi}{3} $.

Step 3: Recall the unit circle values for $ \sin\left(\frac{\pi}{3}\right) $. On the unit circle, $ \frac{\pi}{3} $ corresponds to an angle in the first quadrant, where sine is positive. The sine of $ \frac{\pi}{3} $ is $ \frac{\sqrt{3}}{2} $.

Step 4: Verify the quadrant of the original angle $ \frac{7\pi}{3} $. Since $ \frac{7\pi}{3} > 2\pi $, subtracting $ 2\pi $ places it back in the first quadrant, confirming that the sine value remains positive.

Step 5: Conclude that the exact value of $ \sin\left(\frac{7\pi}{3}\right) $ is $ \frac{\sqrt{3}}{2} $. This matches the sine value of its coterminal angle $ \frac{\pi}{3} $.