How to find cos(2x)

Question

If tan x= 24/7, and 0° < x < 90°, what is cos(2x)?

Answer 1

Hello,

We know that the tangent function represent the ratio of opposite / adjacent, which means if we draw a right triangle with one of the angles as "x", the opposite leg will have a length of 24, and the adjacent leg will have a length of 7. If we then apply the Pythagorean theorem to this triangle, we can find the length of the hypotenuse: 7² + 24² = c² --> c = 25. So we have a right triangle with sides 7, 24, 25.

We are trying to find cos(2x). If we go to a trig identity table we can find an identity that states:

cos(2x) = 2cos²(x) - 1

We need this identity to change the angle of cos(2x) to something with an angle of just "x" because the triangle we found before only has an angle of "x", not "2x". We know that cos represents adjacent/hypotenuse which means for our triangle, cos(x) = 7/25. If we plug in 7/25 for cos(x) into the trig identity we can solve for cos(2x):

cos(2x) = 2cos²(x) - 1

cos(2x) = 2(7/25)² - 1

cos(2x) = 2(49/625) - 1

cos(2x) = 98/625 - 1

cos(2x) = 98/625 - 625/625

cos(2x) = -527/625