Question

Find sin(2x), cos(2x), and tan(2x) from the given information.

Answer 1

If x terminates in the fourth quadrant, then we know that sin(x) < 0 and cos(x) > 0. Given that sec(x) = 8, we immediately have

cos(x) = 1/sec(x) = 1/8

Recall the Pythagorean identity,

cos²(x) + sin²(x) = 1

Then it follows that

sin(x) = -√(1 - cos²(x)) = -3√7/8

Now recall the double angle identities,

sin(2x) = 2 sin(x) cos(x)

cos(2x) = cos²(x) - sin²(x)

Then

sin(2x) = 2 (-3√7/8) (1/8) = -6√7/64 = -3√7/32

cos(2x) = (1/8)² - (-3√7/8)² = -62/64 = -31/32

By definition of tangent,

tan(x) = sin(x)/cos(x)

Then

tan(2x) = sin(2x)/cos(2x) = (=3√7/32) / (-31/32) = 3√7/31