Question

I need someone to tell me how to find the sin of a degree in trigonometry without using the sine, cosine, or tangent buttons on a calculator. Picture included for reference.

Thank you in advance!

Answer 1

For this question, you could have estimated

`\sin31^\circ\approx\sin30^\circ=\frac12`

so that you would expect about

`\sin31^\circ\approx\frac12\approx\frac x7\implies x\approx3.5`

In general, computing
`\sin\theta` (
`\theta` in radians) by hand is tedious and difficult, but there are several ways to do it. Calculutors typically use well-known approximations in the form of truncated series expansions, such as

`\sin\theta\approx\theta-\frac{\theta^3}6+(\theta^5)/(120)`

Then

`\sin31^\circ=\sin(31\pi)/(180)\approx\sin(31\cdot3.14)/(180)\approx0.5148`