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, sin^{2}x + cos^{2}x = 1 , shows the relationship between sin x and cos x. Solve this identity for cos x to obtain
cos x = (+ or ) √ (1  sin^{2}x)

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 sin^{2}x + cos^{2}x = 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 + cot^{2}x = csc^{2}x, we obtain
1 + 1 / tan^{2}x = 1 / sin^{2}x

Solve the above identity for sin x to obtain
sin x = (+ or ) √ [ tan^{2}x / (1 + tan^{2}x) ]

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
Solution to Example 3

First expand the squares.
(sin x + cos x)^{2} + (sin x  cos x)^{2}
= (sin^{2}x + cos^{2}x + 2 cos x sin x) +(sin^{2}x + cos^{2}x  2 cos x sin x)

Group like terms.
= 2 sin^{2}x + 2 cos^{2}x

Factor 2 out
= 2 (sin^{2}x + cos^{2}x)

Use the identity sin^{2}x + cos^{2}x = 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 = 3. Find cos x.

Simplify the trigonometric expression.
(sin x + cos x)(sin x  cos x) + 2 cos^{2}x
Answers to the Above Exercises
 cos(x) =  2√6 / 5 , tan(x) =  √6 / 12
 cos(x) = 3 √(10) / 10
 (sin x + cos x)(sin x  cos x) + 2 cos^{2}x = 1
More References and Links
Trigonometric Identities and Their ApplicationsTrigonometric Formulas and Their Applications