Trigonometric Ratios of `30^\circ`, `45^\circ`, `60^\circ`

In a triangle with angles `30^\circ`, `60^\circ`, `90^\circ`, the lengths of the sides of the triangle are in the ratio:

` 1:2:\sqrt3 `

This ratio has been illustrated in the triangle below:

Example 1:

Find the values of all the six trigonometric ratios of `30^\circ`

Solution: In the triangle shown above, for angle `\theta` = `30^\circ`

Adjacent = ` \sqrt(3)`; Opposite = 1 ; Hypotenuse = 2

Therefore,

cos `30^\circ` = `text(Adjacent)/text(Hypotenuse)` = `\sqrt(3)/(2)` ; sin `30^\circ` = `text(Opposite)/text(Hypotenuse)` = `1/(2)`

tan `30^\circ` = `text(Opposite)/text(Adjacent)` = `1/(\sqrt(3))` ; cosec `30^\circ` = `text(Hypotenuse)/text(Opposite)` = ` 2/(1) `

sec `30^\circ` = `text(Hypotenuse)/text(Adjacent)` = ` 2/(\sqrt(3)) ` ; cot `30^\circ` = `text(Adjacent)/text(Opposite)` = `(\sqrt(3))/(1) `

Example 2:

Find the values of all the six trigonometric ratios of `60^\circ`

Solution: According to the triangle shown below, for angle `\theta` = `60^\circ`

Adjacent = 1 ; Opposite = ` \sqrt(3) ;` Hypotenuse = 2

Therefore,

cos `60^\circ` = `text(Adjacent)/text(Hypotenuse)` = ` 1/(2)` ; sin `60^\circ` = `text(Opposite)/text(Hypotenuse)` = `\sqrt(3)/(2)`

tan `60^\circ` = `text(Opposite)/text(Adjacent)` = `\sqrt(3)/(1)` ; cosec `60^\circ` = `text(Hypotenuse)/text(Opposite)` = `2/(\sqrt(3))`

sec `60^\circ` = `text(Hypotenuse)/text(Adjacent)` = ` 2/(1)` ; cot `60^\circ` = ` text(Adjacent)/text(Opposite)` = `(1)/(\sqrt(3))`

In a triangle with angles `45^\circ`, `45^\circ`, `90^\circ`, the lengths of the sides of the triangle are in the ratio:

`1:1:\sqrt2`

This ratio has been illustrated in the triangle below:

Example 3:

Find the values of all the six trigonometric ratios of `45^\circ`

Solution:In the triangle shown above, for angle `\theta` = `45^\circ`

Adjacent = 1 ; Opposite = 1 ; Hypotenuse = ` \sqrt(2)`

Therefore,

cos `45^\circ` = `text(Adjacent)/text(Hypotenuse)` = `1/(\sqrt(2))` ; sin `45^\circ` = `text(Opposite)/text(Hypotenuse)` = `1/(\sqrt(2))`

tan `45^\circ` = `text(Opposite)/text(Adjacent)` = ` 1/(1)` ; cosec `45^\circ` = `text(Hypotenuse)/text(Opposite)` = `\sqrt(2)/(1)`

sec `45^\circ` = `text(Hypotenuse)/text(Adjacent)` = `\sqrt(2)/(1)` ; cot `45^\circ` = `text(Adjacent)/text(Opposite)` = ` 1/(1)`