I do not understand how to answer this basic trigonometry problem.

Question

If cos(x) = 4/5, cscx < 0, then:
sin2x =
cos2x =
tan2x =

Answer 1

cos(x) = 4/5

cos(x) = adjacent side/hypotenuse

From Pythagorean theorem,

Opposite side = √(hypotenuse² - adjacent side²)

= √(5² - 4²)

= √(25 - 16)

= 3

cos positive and csc is negative in fourth quadrant.

sinx = opposite side/hypotenuse

sinx = - (3/5)

Recall the double - angle identities

sin(2x) = 2 sinx cosx

= 2(-3/5)(4/5)

sin(2x) = - 24/25

 

cos(2x) = cos²x - sin²x

= (4/5)² - (-3/5)²

= (16/25) - (9/25)

cos(2x) =  7/25

 

tan(2x) = sin(2x)/cos(2x)

= [- 24/25]/[7/25]

tan(2x) = - 24/7.