Final answer:
To find the exact value of cos(165°), we use the corresponding angle cos(330°), which is the same as cos(30°), by using the cosine half-angle formula. Since 165° is in the second quadrant where cosine is negative, the exact value is √((2 + √3)/4).
Step-by-step explanation:
The student is asking how to find the exact value of cos(165°) using a half-angle formula. Cosine half-angle formulas allow computation of the cosine of an angle when the cosine of twice that angle is known. The general formula for cosine of a half angle is given by:
cos(θ/2) = ±√((1 + cos(θ))/2)
For cos(165°), we can use the cosine of twice that angle, which is cos(330°). Since 330° is in the fourth quadrant, where cosine is positive, “cos(330°)” is positive and simplifies to:
cos(330°) = cos(360° - 30°) = cos(30°)
Knowing that cos(30°) = √3/2, you can then use the half-angle formula:
cos(165°) = cos(330°/2)
= √((1 + √3/2)/2)
= √((2 + √3)/4)
Since 165° is in the second quadrant and cosine values are negative there, we take the negative value:
cos(165°) = -√((2 + √3)/4)
Therefore, the exact value of cos(165°) is -√((2 + √3)/4).