Find sin x and cos x if x lies in the specified interval

Question

tan x = 3/4, x∈∈ [∏, 3∏ / 2)

Find sin x

Answer 1

The angle is in the 3rd quadrant so both sin x and cos x are negative.

Remember that tan x = (opposite length)/(adjacent length)

Since tan x = 3/4 we can draw a right triangle with side lengths

opposite to angle x has length = 3

adjacent to angle x has length = 4

and the hypotenuse length = √(3² + 4²) = √(9 + 16) = √(25) = 5

We can calculate the absolute value of sin x as

|sin x| = opposite/hypotenuse = 3/5

remembering that sin x is negative in the 3rd quadrant, we get sin x = –3/5

Answer

sin x = –3/5