Final answer:
The exact value of cos(-105°) is found by considering its equivalence to cos(255°), which lies in the third quadrant where cosine is negative. Using the cosine sum identity and the values of sine and cosine at known angles, we can determine that the value is -√3/2.
Step-by-step explanation:
To find the exact value of cos(-105°), we can use the cosine of the related acute angle and consider the cosine function's symmetry properties. The angle -105° is equivalent to 360° - 105°, which is 255°. Since the cosine function is cyclical with a period of 360°, cos(255°) is the same as cos(-105°). We can also express 255° as 180° + 75°, which lies in the third quadrant where cosine values are negative.
Now we can use the cosine sum identity cos(A + B) = cos(A)cos(B) - sin(A)sin(B), where A could be 180° (whose cosine is -1) and B could be 75°. Since sin(75°) = sin(30° + 45°) = sin(30°)cos(45°) + cos(30°)sin(45°) and using the known values sin(30°) = 1/2, cos(30°) = √3/2, sin(45°) and cos(45°) = √2/2, we find that sin(75°) simplifies to (√6 + √2)/4. Therefore, cos(255°) = cos(180° + 75°) = cos(180°)cos(75°) - sin(180°)sin(75°) = (-1)(-√3/2) - (0)(√6 + √2)/4 = √3/2.
Considering the negative value for the third quadrant, the exact value of cos(-105°) is thus -√3/2, which corresponds to option d) √3/2.