Question

Without using the trignometic tables, find i) cos 75°

Answer 1

Notice that 75° = 30° + 45°, so we can rewrite cos(75°) as cos(30° + 45°), and use our angle sum identity.

cos(75°) = cos(30° + 45°) = cos(30°) ⋅ cos(45°) - sin(30°) ⋅ sin(45°)

Now, we just plug in our sine and cosine values for 30° and 45° and simplify.

cos(30°) ⋅ cos(45°) - sin(30°) ⋅ sin(45°)
= (√(3) / 2) ⋅ (√(2) / 2) - (1/2) ⋅ (√(2) / 2)
= √(6) / 4 - √(2) / 4
= (√(6) - √(2)) / 4

We get that the exact value of cos(75°) is (√(6) - √(2)) / 2.