# Using the Trigonometric Identities

Use basic trigonometric identities to find exact values of trigonometric functions and simplify trigonomtric expressions. A tutorial with Several examples including their detailed solutions are presented. A list of the basic Trigonometric Identities is available.

## Examples with Detailed Solutions

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 -) √ (1 - sin2x)

• In quadrant III cos x is negative, hence

cos x = - √ (1 - (-1/3)2)

= - √ (1 - 1/9)

= - √ (8/9) = -(2/3) √ (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 -) √ [ tan2x / (1 + tan2x) ]

• In quadrant IV, sin x is negative. Substitute -5 for tan x, sin x is given by

sin x = - √ [ (-5)2 / (1 + (-5)2) ]

sin x = - √ [ 25 / 26 ] = -5 / √ (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

1. x is in quadrant II and sin x = 1/5. Find cos x and tan x.

2. x is in quadrant I and cot x = 3. Find cos x.

3. Simplify the trigonometric expression.
(sin x + cos x)(sin x - cos x) + 2 cos2x

## Answers to the Above Exercises

1. cos(x) = - 2√6 / 5 , tan(x) = - √6 / 12

2. cos(x) = 3 √(10) / 10

3. (sin x + cos x)(sin x - cos x) + 2 cos2x = 1