Double-Angle Formulas

Question

Find sin 2θ if θ is in the first quadrant and tan θ=40/9.

Answer 1

Using the Pythagorean Theorem and the given value of tan, you can find the hypotenuse, c.

c=√9²+40² = 41

Using this triangle,

sin Θ=40/41

cos Θ=9/41

sin 2Θ = 2sinΘcosΘ

sin 2Θ=(2)(40/41)(9/41) = 720/1681

Answer 2

Let t = θ.

Let r = sqrt(40² + 9²) = sqrt(1681) = 41

Then imagine a right triangle having a base of 9, height of 40, and a hypotenuse of 41.

In this triangle, the angle t is the angle opposite to 40 and adjacent to 9.

Then tan t = 40 / 9, and

sin t = 40 / 41

cos t = 9 / 41

Since sin 2t = 2 (sin t) (cos t)

then sin(2t) = (2) (40/41)(9/41) = 720 / 1681, ans

If we had been told that t was in the third quadrant, then 2t would be in the first quadrant, with the same result.