# Trigonometric Ratios of Complementary Angles

''Two angles whose sum is 90^\circ are called complementary angles.''

Consider the right triangle shown below: In this triangle, \alpha and \beta are complementary angles because:

\alpha+ \beta+90^\circ = 180^\circ (Sum of all the three angles of a triangle is 180^\circ)

=> \alpha+ \beta = 90^\circ

=> \alpha and \beta are complementary angles.

It is customary to represent complementary angles in a right triangle as follows: ## Some Common Complementary Angles:

1. 30^\circ ,60^\circ
2. 45^\circ , 45^\circ

The trigonometric ratios of complementary angles \theta and (90^\circ- \theta) are:

Sin (90^\circ- \theta) = Cos \theta

Cos (90^\circ- \theta) = Sin \theta

Tan (90^\circ- \theta) = Cot \theta

Cosec (90^\circ- \theta) = Sec \theta

Sec (90^\circ- \theta) = Cosec \theta

Cot (90^\circ- \theta) = Tan \theta

##### Example 1:

Given the trigonometric ratios of 30^\circ, find all the six trigonometric ratios of 60^\circ

cos 30^\circ = \sqrt(3 )/( 2 ) ; sin 30^\circ = 1/( 2 )

tan 30^\circ = 1/( \sqrt(3 ) ) ; cosec 30^\circ = 2/( 1 )

sec 30^\circ = 2/( \sqrt(3 ) ) ; cot 30^\circ = ( \sqrt(3 ))/( 1 )

Solution: Since 30^\circ and 60^\circ are complementary angles, we can use the method of co-ratios to determine trigonometric ratios of 60^\circ using those of 30^\circ as shown below:

Sin 60^\circ = Sin (90^\circ- 30^\circ) = Cos 30^\circ = \sqrt(3 )/( 2 )

Cos 60^\circ = Cos (90^\circ- 30^\circ) = Sin 30^\circ = 1/( 2 )

Tan 60^\circ = Tan (90^\circ- 30^\circ) = Cot 30^\circ = ( \sqrt(3 ))/( 1 )

Cosec 60^\circ = Cosec (90^\circ- 30^\circ) = Sec 30^\circ = 2/( \sqrt(3 ) )

Sec 60^\circ = Sec (90^\circ- 30^\circ) = Cosec 30^\circ = 2/( 1 )

Cot 60^\circ = Cot (90^\circ- 30^\circ) = Tan 30^\circ = 1/( \sqrt(3 ) )

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