Question

Given that π < 𝑥 < 2π and 𝑡𝑎𝑛(𝑥) = 3/4, determine the exact value of 𝑐𝑜𝑠2𝑥. Show all work including a diagram, special triangles, CAST rules and related acute angles.

Answer 1

You know that tan θ = y/x, so x = 4 and y = 3
we also know that x^2 + y^2 = r^2
so r = 5
the tangent is positive in I and III, so
in I, tanx = 3/4, sinx = 3/5 and cosx = 4/5
cos 2x = cos^2 x - sin^2 x = 16/25 - 9/25 = 7/25

since we are squaring both sinx and cosx
the same would be true for quad III, so

cos 2x = 7/25

Answer 2

draw a triangle with legs 3 and 4, and you can see that since you are in QIII, that means that
sinx = -3/5 and cosx = -4/5
cos2x = 2cos^2x - 1 = 2 * 16/25 - 1 = 7/25