What is the value of cos 15°?
Given
- cos 15°
- 15° = 45° - 30°
- Identity: cos(A - B) = cosA cosB + sinA sinB
Step 1: Apply the Identity
We know that:
cos 15° = cos(45° - 30°)
Using the identity:
cos(A - B) = cosA cosB + sinA sinB
cos 15° = cos 45° cos 30° + sin 45° sin 30°
Step 2: Substitute Standard Values
- cos 45° = 1/√2
- cos 30° = √3/2
- sin 45° = 1/√2
- sin 30° = 1/2
cos 15° = (1/√2 × √3/2) + (1/√2 × 1/2)
Step 3: Simplify
= √3 / (2√2) + 1 / (2√2)
= (√3 + 1) / (2√2)
Rationalizing the denominator:
cos 15° = (√6 + √2) / 4
Final Answer
The value of cos 15° is (√6 + √2) / 4.