Question

How do you use double angle identity for sin 2x given that cos x= 4/5 where x is an angle in quadrant 1?

Answer 1

See explanation.

Explanation:

The double angle identity lets you calculate the values of trigonometric function of `2x` knowing only the values of trigonometric function of `x`.

In this example you have to calculate `sin2x`. The identity says that:

`sin2x=2sinxcosx`

So first you have to calculate `sinx`. To do this we can use the identity saying that for any real angle `x`:

`sin^2x+cos^2x=1`

If we substitute the given value of `cosx` we get:

`sin^2x+(4/5)^2=1`

`sin^2x=1-16/25`

`sin^2x=9/25`

This equation has 2 solutions `sinx=3/5` or `sin=-3/5`. Using given information that the angle lies in the first quadrant we choose the positive value `sinx=3/5`

Finally we can substitute the values to doble angle formula to calculate:

`sin2x=2*(3/5)*(4/5)`

`sin2x=24/25`