Calculate sin 2x if tan x=3 and x is in (0 , 90)?
Solution
Since x is in the first quadrant, then the values of the trigonometric functions are positive.
We'll write the formula for sin 2x:
sin 2x = sin (x+x)
sin (x+x) = sinx*cosx + sinx*cosx
sin (x+x) = 2 sinx*cos x
We know that the tangent function is a ratio of sine and cosine functions:
tan x = sin x/cos x
tan x = 3 => sin x/cos x = 3 => sin x = 3 cos x
sin 2x = 2*3cos x*cosx = 6 (cos x)^2
We know that:
1 + (tan x)^2 = 1/(cos x)^2
1 + 9 = 1/(cos x)^2
10 = 1/(cos x)^2
(cos x)^2 = 1/10
sin 2x = 6/10
sin 2x = 3/5