How do you find sin 2x, given tan x = -2 and cos x > 0?

1 Answer
Nghi N.
Nov 30, 2015

Find sin 2x, knowing tan x = -2, and cos x > 0

Ans: #sin 2x = 4/5#

Explanation:

3 Trig identities to be used:
#1 + tan^2 x = 1/cos^2 x#(1)
#sin^2 x + cos ^2 x = 1# (2)
#sin 2x = 2sin x.cos x# (3)
Given tan x = -2. First find cos x and sin x
(1) --> #1 + 4 = 1/(cos^2 x)# --> #cos^2 x = 1/5# --> #cos x = +- 1/sqrt5#.
Since cos x > 0, then #cos x = 1/sqrt5.#
(2) --> #sin^2 x = 1 - cos^2 x = 1 - 1/5 = 4/5# --># sin x = +- 2/sqrt5.#
Since cos x > 0 then #sin x = 2/sqrt5#.
(3) --> #sin 2x = 2sin x.cos x = 2(1/sqrt5)(2/sqrt5) = 4/5#

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