Question

How do you use the double-angle identities to find sin(2x) if csc x= -root21 and cos x is less than 0?

Answer 1

Find sin 2x knowing `csc x = -sqrt21`

Ans: `((4sqrt5)/21)`

Explanation:

`csc x = 1/(sin x) = - sqrt21`
`sin x = - 1/(sqrt21)`
`cos^2 x = 1 - sin^2 x = 1 - 1/21 = 20/21`
`cos x = - (2sqrt5)/sqrt21` (cos x less than 0)
`sin 2x = 2sin x.cos x = 2(-1/(sqrt21))((-2sqrt5)/(sqrt21)) = ((4sqrt5)/21)`

Check by calculator.
sin x = -1/sqrt21 = -0.22 --> x = -167.29 --> 2x = --> sin (-334.58) = 0.43
`(4sqrt5)/21 = 8.94/21 = 0.43.` OK