# Sin(x)=5/13 and x is in quadrant 1: find sin 2x, cos 2x, and tan 2x.?

I know this is trigonometric double angle formula, but beyond that I’m stumped.

• Identities needed:

1) sin 2x = 2 sin x cos x

2) cos 2x = 2cos^2 x -1

3) cos x = √1-sin^2 x

4) tan x = sin x / cos x

since sin x = 5/13, using identity (3) gives:

cos x = √(1-25/169) = 12/13

Using identity (2), it can be found that:

>>>cos 2x = 2*(12/13)^2 -1

using identity (1):

>>>sin 2x = 2* 5/13 * 12/13 = 120/169

using identity (4):

>>>tan 2x = [120/169] / [2*(12/13)^2 -1]

• If sin x = 5/13 then cos x= 12/13

Now, just apply the formulae and substitute the values of Sin x and Cos x

Sin 2x = 2*Sin x*Cos x= 120/169

Cos 2x = Cos^2 x – Sin^2 x = 144/169 – 25/169 =119/169

Tan 2x = Sin 2x/Cos 2x = 120/119.

• cos x = 12/13

sin 2x = 2 sin x cos x = 2 x 5/13 x 12/13 = 120/169

cos 2x = 2 cos ² x – 1 = 119/169

tan 2x = 120/119

• You’ll also need cos(x), which you can get by using sin^2(x) + cos^2(x) = 1. Solve for cos(x). You’ll have a choice of a positive or negative answer. The info that it’s in quadrant 1 tells you whether the cosine should be positive or negative.

Once you have those, plug them into the formulas for sin(2x) and cos(2x), which are in terms of sin(x) and cos(x).

And then tan(2x) = sin(2x)/cos(2x).