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:
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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)
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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
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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
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Solve the above identity for sin x to obtain
sin x = (+ or -) √ [ tan2x / (1 + tan2x) ]
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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
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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)
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Group like terms.
= 2 sin2x + 2 cos2x
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Factor 2 out
= 2 (sin2x + cos2x)
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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 = 3. Find cos x.
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Simplify the trigonometric expression.
(sin x + cos x)(sin x - cos x) + 2 cos2x
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 cos2x = 1
More References and Links
Trigonometric Identities and Their ApplicationsTrigonometric Formulas and Their Applications