Question

What is value of sin theta where theta is 35

Answer 1

Start by observing that sin 35° is close to sin 30°, which is 1/2. So we immediately know it is roughly 1/2. That's within about 7% of the actual value.

Let's try to get a better estimate. By the angle addition identity,

sin 35° = sin 30° cos 5° + sin 5° cos 30° = (1/2) cos 5° + sin 5° 3–√/23/2.

Now, since 5° = π/36 is a relatively small angle, we can use the approximations sin x ≈ x and cos x ≈ 1. So

sin 35° ≈ 1/2 + (π/36)(3–√/23/2).

Now π ≈ 22/7, and 3–√3 ≈ 7/4 because 49/16 ≈ 3. So we get

sin 35° ≈ 1/2 + (22/7)(1/36)(1/2)(7/4) = 1/2 + 11/144 = 83/144,

This differs from the true value by less 1%.