3. Unit Circle

Reference Angles

3. Unit Circle

Reference Angles

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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

First, understand that coterminal angles are angles that share the same terminal side. To find a coterminal angle, you can add or subtract multiples of 2π from the given angle.

Given the angle $ \frac{7\pi}{3} $, we need to find a coterminal angle between 0 and 2π. Start by subtracting 2π from $ \frac{7\pi}{3} $ to bring it within the desired range.

Convert 2π to a fraction with a denominator of 3: $ 2\pi = \frac{6\pi}{3} $. Subtract this from $ \frac{7\pi}{3} $ to find the coterminal angle: $ \frac{7\pi}{3} - \frac{6\pi}{3} = \frac{\pi}{3} $.

Now, evaluate $ \sin\left(\frac{\pi}{3}\right) $. Recall that $ \sin\left(\frac{\pi}{3}\right) $ is a standard angle in the unit circle, and its value is $ \frac{\sqrt{3}}{2} $.

Thus, the exact value of $ \sin\left(\frac{7\pi}{3}\right) $ is $ \frac{\sqrt{3}}{2} $, as it is coterminal with $ \frac{\pi}{3} $.

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For each expression, identify which coterminal angle to use & determine the exact value of the expression.sin7π3\sin\frac{7\pi}{3}sin37π

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Reference Angles practice set

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