Question

If cos alpha = - 5/13 and tan a < 0 , then find sin 2 alpha

Answer 1

Recall the Pythagorean identity,

cos²(x) + sin²(x) = 1

and the definition of tangent,

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

Given that tan(α) < 0 and cos(α) = -5/13 < 0, it follows that sin(α) > 0. Then using the identity above, we find

sin(α) = √(1 - cos²(α)) = 12/13

Now recall the double angle identity for sine,

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

Then

sin(2α) = 2 sin(α) cos(α) = 2 • 12/13 • (-5/13) = -360/169