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, 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 -) square root (1 - sin^{2}x)

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 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 -) square root [ 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 = - 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}

= (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.