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
  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 = 1/5. Find cos x.

  3. 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