Find sin 2x, cos 2x, and tan 2x?

Question

If tan x = -1/3, cos x >0, then 
sin 2x=?
cos 2x=?
tan 2x=?

Answer 1

Given that tanx = -1/3

Asked to prove sin2x  , cos2x and tan2x

In a triangle

Opposite side = -1 and adjcent side = 3 and hypotenuse = a

According to pythagrerous theroem

sin2x = 2 sinx cosx

Therefore the exact values of sin2x and cos2x and tan2x

Answer 2

tan x = - 1/3.

Double angle formulas : sin 2x = 2 tan x / (1 + tan² x),

                                    cos 2x = (1 - tan² x)/(1 + tan² x), and

                                     tan 2x = 2 tan x / (1 - tan² x).

sin 2x = 2 tan x / (1 + tan² x)

= 2(- 1/3) / (1 + (- 1/3)²)

= (- 2/3) / (10/9)

= (- 2 * 3)/10

= - 3/5.

cos 2x = (1 - tan² x)/(1 + tan² x)

= (1 - (- 1/3)²) / (1 + (- 1/3)²)

= (1 - 1/9) / (1 + 1/9)

= 8/10

= 4/5.

tan 2x = 2 tan x / (1 - tan² x)

= 2(- 1/3) / (1 - (- 1/3)²)

= (- 2/3) / (8/9)

= (- 2 * 3)/8

= - 3/4.

Therefore, sin 2x = - 3/5, cos 2x = 4/5, and tan 2x = - 3/4.