Final answer:
The exact value of cos 285° is found by using sum identities and breaking down the angle into known angles, resulting in an exact value of (sqrt(6) - sqrt(2))/4.
Step-by-step explanation:
To find the exact value of cos 285° we can use the sum identity for cosine. The sum identity for cosine is given by cos(A + B) = cos A cos B - sin A sin B. The angle 285° can be expressed as a sum of two angles, for instance, 360° - 75°, which is a full rotation minus an angle whose cosine we know.
Now, let's use the identity with A = 360° and B = -75°:
cos(285°) = cos(360° - 75°) = cos 360° cos 75° + sin 360° sin 75°. Since cos 360° = 1 and sin 360° = 0, this simplifies to cos 285° = cos 75°.
To find cos 75°, we can further break it down using angle sum identities to known angles, such as 30° and 45°, where cos(75°) = cos(30° + 45°). Using the sum identity again, we get cos(30° + 45°) = cos 30° cos 45° - sin 30° sin 45°, which using known values results in √3/2 * √2/2 - 1/2 * √2/2.
Hence, the exact value of cos 285° is √3/2 * √2/2 - 1/2 * √2/2 which simplifies to (√6 - √2)/4.