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

1 Answer
Oct 7, 2015

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