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

Verified step by step guidance

1. Recognize that 195° is not a standard angle, but it can be expressed in terms of an angle whose cosine value is known. Notice that 195° = 2 × 97.5°, so we can use the half-angle identity for cosine by setting \( \theta = 195° \) and \( \frac{\theta}{2} = 97.5° \). However, since 97.5° is not a standard angle either, let's try expressing 195° as \( 180° + 15° \) to use angle sum identities or consider the half-angle identity for \( 195° = 2 \times 97.5° \) directly.
2. Recall the half-angle identity for cosine: \[ \cos \left( \frac{\theta}{2} \right) = \pm \sqrt{ \frac{1 + \cos \theta}{2} } \] Since we want \( \cos 195° \), we can rewrite it as \( \cos (2 \times 97.5°) \) and then use the double-angle formula or use the half-angle identity by setting \( \theta = 390° \) (which is \( 2 \times 195° \)) and then find \( \cos 195° = \cos \left( \frac{\theta}{2} \right) \). But this might be complicated, so instead, let's use the half-angle identity by expressing 195° as \( 180° + 15° \) and then use the cosine addition formula or the half-angle identity for 15°.
3. Alternatively, express 195° as \( 180° + 15° \). Since cosine has the property: \[ \cos (180° + x) = -\cos x \] we can write: \[ \cos 195° = \cos (180° + 15°) = -\cos 15° \] Now, focus on finding \( \cos 15° \) using the half-angle identity.
4. To find \( \cos 15° \), use the half-angle identity with \( \theta = 30° \) because \( 15° = \frac{30°}{2} \). Apply the half-angle formula: \[ \cos 15° = \cos \left( \frac{30°}{2} \right) = \pm \sqrt{ \frac{1 + \cos 30°}{2} } \] Determine the sign by considering the quadrant of 15°, which is in the first quadrant where cosine is positive.
5. Substitute the known value \( \cos 30° = \frac{\sqrt{3}}{2} \) into the formula: \[ \cos 15° = \sqrt{ \frac{1 + \frac{\sqrt{3}}{2}}{2} } \] Simplify the expression inside the square root to get the exact value of \( \cos 15° \). Finally, recall from step 3 that \( \cos 195° = -\cos 15° \), so multiply your result by -1 to find the exact value of \( \cos 195° \).

Key Concepts

Half-Angle Identities

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

Reference Angles and Quadrants

Understanding the quadrant in which the angle lies is crucial to determine the sign of the trigonometric value. Since 195° is in the third quadrant, and half of 195° is 97.5°, which lies in the second quadrant, the sign of cos(97.5°) is negative. This guides the correct choice of the ± sign in the half-angle formula.

Exact Values of Cosine for Common Angles

To use the half-angle identity, you need the exact cosine value of the original angle, often a multiple of 30°, 45°, or 60°. For 195°, the related angle is 390° (2 × 195°), but since 195° = 360° - 165°, you can use cos 165° = -cos 15°, and cos 15° can be found using sum or difference formulas. Knowing these exact values is essential for precise calculation.