Question

If sin(x) = 0.4 , what is sin((3π/2)+x)=? , Show exact values. How would I get along solving this without a calculator.

Answer 1

`- 0.916`

Explanation:

Use trig identity:
`sin (a + b) = sin a*cos b + sin a*sin b`

In this case:

`sin ((3pi/2) + x) = sin ((3pi)/2)*cos x + cos ((3pi)/2)*sin x`

Since `sin ((3pi)/2) = - 1` and `cos ((3pi)/2) = 0`, therefore:
`sin ((3pi)/2 + x) = - cos x`.

Find `cos x`, knowing `sin x = 0.4.`

`cos^2 x = 1 - sin^2 x = 1 - 0.16 = 0.84`

Recall that `0 < sinx < 1/2pi therefore cosx > 0`

`cos x = sqrt(0.84) = 0.916`

`sin ((3pi)/2 + x) = +- 0.916`