Use a half-angle identity to find each exact value.
sin 165°

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Recognize that 165° can be expressed as half of 330°, so we can use the half-angle identity for sine: \(\sin\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos\theta}{2}}\).

Set \(\theta = 330^\circ\), so \(\sin 165^\circ = \sin\left(\frac{330^\circ}{2}\right)\).

Determine the sign of \(\sin 165^\circ\). Since 165° is in the second quadrant where sine is positive, we take the positive root.

Calculate \(\cos 330^\circ\) using the unit circle or cosine values: \(\cos 330^\circ = \cos(360^\circ - 30^\circ) = \cos 30^\circ\) but positive or negative? Recall cosine is positive in the fourth quadrant, so \(\cos 330^\circ = \sqrt{3}/2\).

Substitute \(\cos 330^\circ\) into the half-angle formula: \(\sin 165^\circ = + \sqrt{\frac{1 - \frac{\sqrt{3}}{2}}{2}}\). This expression gives the exact value of \(\sin 165^\circ\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Half-Angle Identities

Half-angle identities express the sine, cosine, or tangent of half an angle in terms of the cosine of the original angle. For sine, the identity is sin(θ/2) = ±√[(1 - cos θ)/2]. The sign depends on the quadrant of θ/2. These identities help find exact trigonometric values for angles not commonly found on the unit circle.

Reference Angles and Quadrants

Understanding the quadrant in which an angle lies is crucial for determining the sign of trigonometric functions. Since 165° is in the second quadrant, and half of 165° is 82.5°, which is also in the first quadrant, the sine value will be positive. This knowledge ensures correct application of the half-angle formula.

Exact Values of Cosine for Special Angles

To use the half-angle identity for sin 165°, you need the exact value of cos 330° (since 165° = 330°/2). Knowing exact cosine values for special angles like 30°, 45°, 60°, and their multiples allows precise calculation without approximations. This forms the basis for deriving exact sine values using half-angle formulas.

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