Given sec𝑥 = 29/16 in quadrant I, determine the exact value of sin2𝑥.

Answer 1

First, recall that sec(x) = 1 / cos(x). We know that sec(x) = 29/16, therefore cos(x) = 16/29.

Use this information to sketch a right triangle with adjacent side length 16, and hypotenuse length 29. (Think about SohCahToa).

Now that we have two sides of a right triangle, we can solve for the final side length (the opposite side), using the Pythagorean theorem.

a² + 16² = 29²

a² = 585

a = √585

Now that we have a triangle sketched with all 3 side lengths, let's consider what we are trying to solve for, sin(2x).

Recall the double angle identity for sine. sin(2x) = 2sin(x)cos(x)

We can use the triangle that we sketched to find the values of sin(x) and cos(x). So,

sin(2x) = 2(√585 / 29)(16 / 29) = 32√585 / 841 = 96√65 / 841