Use a half-angle identity to find each exact value.
cos 195°

Verified step by step guidance

1. <Step 1: Identify the half-angle identity for cosine.> \( \cos(\theta) = \cos(2\alpha) = 2\cos^2(\alpha) - 1 \) can be rearranged to find \( \cos(\alpha) \) using the half-angle identity: \( \cos(\alpha) = \pm \sqrt{\frac{1 + \cos(2\alpha)}{2}} \).
2. <Step 2: Express 195° as a half-angle.> Notice that 195° can be expressed as 390°/2. Therefore, \( \alpha = 195° \) and \( 2\alpha = 390° \).
3. <Step 3: Find \( \cos(390°) \).> Since 390° is equivalent to 30° (because 390° - 360° = 30°), we have \( \cos(390°) = \cos(30°) = \frac{\sqrt{3}}{2} \).
4. <Step 4: Substitute \( \cos(390°) \) into the half-angle identity.> Substitute \( \cos(390°) = \frac{\sqrt{3}}{2} \) into the half-angle formula: \( \cos(195°) = \pm \sqrt{\frac{1 + \frac{\sqrt{3}}{2}}{2}} \).
5. <Step 5: Determine the correct sign for \( \cos(195°) \).> Since 195° is in the third quadrant where cosine is negative, choose the negative sign: \( \cos(195°) = -\sqrt{\frac{1 + \frac{\sqrt{3}}{2}}{2}} \).

Key Concepts

Half-Angle Identities

Half-angle identities are trigonometric formulas that express the sine and cosine of half an angle in terms of the sine and cosine of the original angle. For cosine, the identity is cos(θ/2) = ±√((1 + cos(θ))/2). These identities are particularly useful for simplifying expressions and finding exact values of trigonometric functions at specific angles.

Reference Angles

A reference angle is the acute angle formed by the terminal side of an angle and the x-axis. For angles greater than 180°, like 195°, the reference angle helps determine the corresponding angle in the first quadrant, which is essential for evaluating trigonometric functions. In this case, the reference angle for 195° is 195° - 180° = 15°.

Quadrant Analysis

Quadrant analysis involves understanding the signs of trigonometric functions based on the quadrant in which the angle lies. The angle 195° is in the third quadrant, where cosine values are negative. This knowledge is crucial when applying half-angle identities, as it affects the sign of the resulting value when calculating cos(195°) using the half-angle identity.