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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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.

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Double Angle Identities

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.

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Reference Angles on the Unit Circle

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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