How do you use double-angle identities to find the exact value of sin 2x & cos 2x when csc(x) = -25/7 and cos x>0?

1 Answer
Apr 23, 2018

Find sinx first and then find sin2x and cos2x.

Explanation:

As cos x > 0, the basic angle must be in quadrant II or III.
As csc x = -25/7, the basic angle must be in quadrant III as csc x = 1/sin x.

By algebra, sin x = -7/25. From there, you can draw the triangle and find out what the basic angle is.
enter image source here

sin 2x = 2sin x cos x
cos 2x = 1- 2sin^2x (there are other variations but this works better for this question)

From the triangle, we can find that cos x = -24/25
The negative sign is because the basic angle is in the third quadrant, where only tangent is positive,

Evaluating sin 2x and cos 2x, sin2x = -336/625 and cos2x = -527/625.

The negative signs are because the angles fall in the third quadrant. Not so sure about that though.