Final answer:
To calculate cos(165°), use the cosine addition formula by expressing 165° as the sum of 120° and 45°. This yields cos(165°) = -(√2 + √6)/4.
Step-by-step explanation:
To calculate cos(165°), use the cosine addition formula by expressing 165° as the sum of 120° and 45°. This yields cos(165°) = -(√2 + √6)/4.
To find the exact value of the expression cos(165°), we can use the cosine addition formula, which is cos(A + B) = cos(A)cos(B) - sin(A)sin(B). We can express 165° as the sum of two angles whose cosine and sine values we know from the unit circle, such as 120° and 45°. Therefore, cos(165°) = cos(120° + 45°).
Using the addition formula, we get: cos(165°) = cos(120°)cos(45°) - sin(120°)sin(45°). The cos(120°) is -1/2, sin(120°) is √3/2, and both cos(45°) and sin(45°) are √2/2. Substituting these values in, we get cos(165°) = (-1/2)(√2/2) - (√3/2)(√2/2) which simplifies to -√2/4 - √6/4, or as a single fraction -(√2 + √6)/4.