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

Verified step by step guidance

1. Recognize that 195° can be expressed as 390°/2, which allows us to use the half-angle identity for sine.
2. Recall the half-angle identity for sine: \( \sin\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos(\theta)}{2}} \).
3. Determine \( \theta \) such that \( \theta = 390° \) because \( 195° = \frac{390°}{2} \).
4. Calculate \( \cos(390°) \) using the fact that \( 390° = 360° + 30° \), so \( \cos(390°) = \cos(30°) \).
5. Substitute \( \cos(390°) \) into the half-angle identity and simplify to find \( \sin(195°) \).

Key Concepts

Half-Angle Identities

Half-angle identities are trigonometric formulas that express the sine, cosine, and tangent of half an angle in terms of the trigonometric functions of the original angle. For sine, the identity is sin(θ/2) = ±√((1 - cos(θ))/2). These identities are particularly useful for finding exact values of trigonometric functions at angles that are not standard, such as 195°.

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°, the reference angle helps in determining the sine and cosine values by relating them to their corresponding acute angles. For sin(195°), the reference angle is 195° - 180° = 15°, which is essential for calculating the sine value using the half-angle identity.

Quadrant Analysis

Quadrant analysis involves determining the sign of trigonometric functions based on the quadrant in which the angle lies. The angle 195° is in the third quadrant, where sine values are negative. Understanding the quadrant helps in applying the half-angle identity correctly, ensuring that the final value of sin(195°) reflects the appropriate sign.