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:
- `30^\circ` ,`60^\circ`
- `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 ) )`