Finding exact values for sin(2x), cos(2x), and tan(2x)

Question

Find the exact values of ,  and  without solving for .If , and  is in quadrant .

Answer 1

Hey! Lets solve this shall we ?

So, we must find the three Double-Angle Identities sin2x, cos2x and tan2x. If tanx = -4, then cotx = -1/4 from taking the reciprocal of both sides. Then Pythagorean Theorem says (-1)² + (4)² = z². Then z = √17 which is the hypotenuse. Now, that makes sinx = 4√17/17 and cosx = -√17/17. Lets solve for sin2x first such that

sin2x = 2sinxcosx = 2(4√17/17)(-√17/17) = -8(17) / [(17)(17)] = -8/17. Now, lets solve for cos2x such that

cos2x = cos²x - sin²x = (-√17/17)² - (4√17/17)² = (17/289) - (16*17/289) = (1/17) - (16/17) = -15/17. Finally, we solve for tan2x, where tanx = -4 such that

tan2x = (2tanx) / (1-tan²x) = [2(-4)] / [1-[-4]²] = -8 / [1-16] = -8 / -15 = 8/15

I hope this helped!