Finding sin x and cos x when tan 2x = -24/7

[glass.shards]

If tan2x=(-24/7), find the exact value(s) of sinx and cosxWorking out the answers by hand, I get

sinx = ±3/5, ±4/5
cosx = ±3/5, ±4/5But by actually calculating x and plugging it into sinx and cosx, I get

sinx = 3/5, 4/5
cosx = 3/5, -4/5I'm pretty sure that the latter are the answers, but how do I justify it given the ±?

 

[glass.shards]

Here's what I did:

tan2x = -24/7

so to find cosx and sinx, I established that on a graph:
y = -24
x = 7
r = 25

OR

y = 24
x = -7
r = 25

Thus, cos2x = ±7/25. By breaking down cos2x into (1-2(sinx)^2) and (2(cosx)^2-1) and working out the answers, I got those four values.

 

[Gib Z]

You get extra false solutions from taking both positive and negative values from the square root. The only way to justify the latter answers is to verify each of the possibilities and saying some did not evaluate to the correct value.