Find the exact value of cos 75 degrees in terms of radicals. * (
)/4 (
)/4 (
)/4 None of the above
The Correct Answer and Explanation is:
Correct Answer: The correct option is (√6 – √2)/4.
Explanation:
To find the exact value of cos(75°) in terms of radicals, we can express 75° as a sum of two special angles for which the exact trigonometric values are well-known. The most convenient way to do this is to write 75° as the sum of 45° and 30°.
1. Decompose the Angle:
We can write 75° as:
75° = 45° + 30°
2. Apply the Cosine Angle Addition Formula:
The angle addition formula for cosine is:
cos(A + B) = cos(A)cos(B) – sin(A)sin(B)
By substituting A = 45° and B = 30° into this formula, we get:
cos(75°) = cos(45° + 30°) = cos(45°)cos(30°) – sin(45°)sin(30°)
3. Substitute the Known Values of Special Angles:
We know the exact values for the sine and cosine of 45° and 30°:
- cos(45°) = √2 / 2
- sin(45°) = √2 / 2
- cos(30°) = √3 / 2
- sin(30°) = 1 / 2
Now, we substitute these values into our equation:
cos(75°) = (√2 / 2) * (√3 / 2) – (√2 / 2) * (1 / 2)
4. Simplify the Expression:
First, multiply the terms:
cos(75°) = (√2 * √3) / (2 * 2) – (√2 * 1) / (2 * 2)
cos(75°) = √6 / 4 – √2 / 4
Next, combine the terms over the common denominator:
cos(75°) = (√6 – √2) / 4
This result matches the third option provided in the question. Since 75° is in the first quadrant, its cosine value must be positive. As √6 ≈ 2.449 and √2 ≈ 1.414, the numerator (√6 – √2) is positive, which is consistent with our expectation.thumb_upthumb_down
