Trigonometric Ratios of Complementary Angles

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

Consider the right triangle shown below:

Right Triangle

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:

represent-complementary

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 ) )`






Become a member today (it’s Free)!


 Register with us

Are you a member? Sign in!


 Login to your account