Final answer:
The exact value of cos (15°) can be found using half-angle formulas, and since it is in the first quadrant where cosine is positive, the answer is √(2 + √3)/2.
Step-by-step explanation:
To provide an exact evaluation of cos (15°), we can use the half-angle formulas. Recall that the cosine of a double angle (θ) is represented as: cos(2θ) = cos²(θ) - sin²(θ) and that can also be expressed as cos(2θ) = 2cos²(θ) - 1 or cos(2θ) = 1 - 2sin²(θ). We will apply the second version to cos (30°), which is a known value.
cos (30°) is equal to √3/2. And since 15° is half of 30°, we need to find the cosine of half of 30°. We rearrange the double angle formula to solve for cos(θ): cos²(θ) = (1 + cos(2θ))/2. So cos²(15°) = (1 + cos(30°))/2 = (1 + √3/2)/2.
Therefore, cos (15°) = ±√((1 + √3/2)/2). Since 15° is in the first quadrant where cosine values are positive, we take the positive root.
±√((2 + √3)/4) = √(2 + √3)/2. Hence, cos (15°) = √(2 + √3)/2.