# Using the Trigonometric Identities

How basic trigonometric identities are used? A tutorial with Several examples with detailed solutions are presented. A list of the basic Trigonometric Identities is available.

 Example 1: x is an angle in quadrant III and sin x = -1 / 3. Find cos x. Solution to Example 1: One of the Pythagoren identities, sin2x + cos2x = 1 , shows the relationship between sin x and cos x. Solve this identity for cos x to obtain cos x = (+ or -) square root (1 - sin2x) In quadrant III cos x is negative, hence cos x = - square root (1 - (-1/3)2) = - square root (1 - 1/9) = - square root (8/9) = -(2/3)square root(2) As an exercise, use the value of sin x given above and cos x found to check that sin2x + cos2x = 1. Example 2: x is an angle in quadrant IV and tan x = -5. Find sin x. Solution to Example 2 Using the reciprocal identities csc x = 1 / sin x and cot x = 1 / tan x in the Pythagorean identity 1 + cot2x = csc2x, we obtain 1 + 1 / tan2x = 1 / sin2x Solve the above identity for sin x to obtain sin x = (+ or -) square root [ tan2x / (1 + tan2x) ] In quadrant IV, sin x is negative. Substitute -5 for tan x, sin x is given by sin x = - square root [ (-5)2 / (1 + (-5)2) ] sin x = - square root [ 25 / 26 ] = -5 / square root (26) Example 3: Simplify the trigonometric expression (sin x + cos x)2 + (sin x - cos x)2 Solution to Example 3 First expand the squares. (sin x + cos x)2 + (sin x - cos x)2 = (sin2x + cos2x + 2 cos x sin x) +(sin2x + cos2x - 2 cos x sin x) Group like terms. = 2 sin2x + 2 cos2x Factor 2 out = 2 (sin2x + cos2x) Use the identity sin2x + cos2x = 1 to simplify the above expression. = 2 Exercises x is in quadrant II and sin x = 1/5. Find cos x and tan x. x is in quadrant I and cot x = 1/5. Find cos x. Simplify the trigonometric expression. (sin x + cos x)(sin x - cos x) + 2 cos2x Trigonometric Identities and Their Applications and Trigonometric Formulas and Their Applications