if cos x = 1/4 , and x is in quadrant 4

Question

Find exact values of

1) sin(2x)

2)cos(2x)

3)tan2(x)

Answer 1

Hey! Lets solve this shall we ?

So, we must find the Double-Angle Identities of sin2x, cos2x and tan2x as follows, given that cosx = 1/4. If cosx = 1/4, then sinx = √15/4 after applying the Pythagorean Theorem. Lets begin by finding sin2x where

sin2x = 2sinxcosx = 2(√15/4)(1/4) = 2√15/16 = √15/8

Now, lets find cos2x such that cos2x = cos²x - sin²x and plugging everything in to obtain

cos2x = (1/4)² - (√15/4)² = (1/16) - (15/16) = -14/16 = -7/8

Then, we can find tan2x because tanx = (sinx)/(cosx) = (√15/4)/(1/4) = (√15/4)*(4/1) = √15. Thus,

tan2x = (2tanx) / (1-tan²x) = (2√15) / (1-(√15)²) = 2√15 / (1-15) = (2√15) / -14 = -√15/7

I hope this helped!