double angle formula?

Question

How can i find the exact values of sin2x and cos2x when:

sinx = -7/15, pi
help!

Answer 1

You're given that

sin(x) = -7/15, and note that sine is negative in quadrants 3 and 4.

You're also given pi < x < 3pi/2, which occurs in quadrants 2 and 3.

Therefore, your answer will be in quadrant 3.

Your first step would be to solve for cos(x). To do that, make a right angle triangle with angle x. You're going to have
1) an opposite side (i.e the side angle x is projecting on,
2) a hypotenuse (the longest side)
3) an adjacent side (the remaining side).

sin(x) = -7/15, but ignore the minus sign for now, and say
sin(x) = 7/15. Remember the rules of SOHCAHTOA? It says
sine is equal to opposite over hypotenuse.

sin(x) = opp/hyp = 7/15

Make the triangle have its opposite side 7, and hypotenuse 15.
It follows that the adjacent side would be sqrt(15^2 - 7^2), or
sqrt(225 - 49) = sqrt(176).

Therefore, opp = 7, hyp = 15, adj = sqrt(176)

sin(x) = opp/hyp = 7/15
cos(x) = adj/hyp = sqrt(176)/15

However, since we're in quadrant 3, your answer will actually be negative.

cos(x) = -sqrt(176)/15

To find the exact values of sin(2x) and cos(2x), we'll first state what they're equal to. The double angle identity states the following:

sin(2x) = 2sin(x)cos(x)

Plugging in sin(x) = (-7/15), cos(x) = -sqrt(176)/15, we have

sin(2x) = 2(-7/15)(-sqrt(176)/15)
sin(2x) = 49sqrt(176)/225

Also by the double angle identity,
cos(2x) = cos^2(x) - sin^2(x), so

cos(2x) = [-sqrt(176)/15]^2 - [-7/15]^2
cos(2x) = 176/225 - 49/225
cos(2x) = 127/225

Answer 2

If sinx = -7/15 , then cosx = - 4sqrt(11)/15
sin2x = 2sinxcosx
cos 2x = 1-2sin^2x
So just plug in the numbers and you've got it made.