Question

tan(theta)=(7/24), sin(theta)<0
find cos(2theta)

Answer 1

If tan is positive and sin is negative, theta is in the third quadrant. From Pythagorean considerations, sin theta = -7/25 and cos theta = -24/25. (Imagine a 7,25,25 right triangle)

Use the double angle formula for
cos(2 theta)
= cos^2(theta) - sin^2(theta)
= (24^2-7^2)/25^2 = 527/625
2theta will be in the first quadrant, 32.5 degrees

Check: theta = 180 + 16.26 = 196.26 deg
2 theta = 392.50 => 32.5 degrees

Answer 2

Change the third sentence to:
(Imagine a 7,24,25 right triangle)