How to find sin 1degree without using a calculator

Question

Kayla claims that she can find an approximate value for sin 1o without using a machine to do any
computations. Explain or show how she can do this.

Answer 1

You can use the infinite power series expansion of sin x.

 

sin x = x - x³/3! + x⁵/5! - x⁷/7! + ...

 

where x is in radians.  If x is very small (x << 1 radian), you get a good approximation for sin x from

 

sin x ≅ x

 

(By the way, this approximation is used frequently in optics and quantum physics.)

 

In this problem, x = 1° = 1°(π/180) radians ≅ 0.017453292

 

sin(1º) ≅ 0.017452406

 

Note x and sin x are close for this very small value of x.  If x is not very small, just include more terms from the series.