Given sin x = 4/5 and x in Q1, find cos 2x.

Question

Given sin x = 4/5 and x in Q1, find cos 2x.

Answer 1

We can derive the double angle formula for cosine by taking the addition formula:

cos(x + y) = cos(x)cos(y) - sin(x)sin(y)

and replacing y for x:

cos(x + x) = cos(2x) = cos(x)cos(x) - sin(x)sin(x) = cos²(x) - sin²(x)

We can put this entirely in terms of sin(x) by recognizing that cos²(x) = 1 - sin²(x), allowing us to rewrite the double angle formula for cosine as

cos(2x) = (1 - sin²(x)) - sin²(x) = 1 - 2sin²(x)

Using sin(x) = 4/5, we can now solve for cos(2x):

cos(2x) = 1 - 2(4/5)²

cos(2x) = 1 - 2(16/25)

cos(2x) = 1 - (32/25)

cos(2x) = 25/25 - 32/25

cos(2x) = -7/25