# 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)

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

The table given below summarizes the trigonometric ratios of angles 0^\circ, 90^\circ, 180^\circ, 270^\circ

 \theta Cos \theta Sin \theta Tan \theta Cosec \theta Sec \theta Cot \theta 30^\circ \sqrt(3)/(2) 1/(2) 1/(\sqrt(3)) 2/(1) 2/\sqrt(3) (\sqrt(3))/(1) 45^\circ 1/(\sqrt(2)) 1/(\sqrt(2)) 1 \sqrt2 \sqrt2 1 60^\circ 1/(2) \sqrt(3)/(2) \sqrt(3)/(1) 2/\sqrt(3) 2/(1) 1/\sqrt(3)

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