# Cosecant: in a Right Triangle

How to find cosecant in a right triangle (trigonometry): formula, 1 example, and its solution.

## Formula

Cosecant is the reciprocal of sine.

So, to find cosecant (csc A),

first write 1/[sin A],

find sin A = (Opposite side)/(Hypotenuse),

and write the reciprocal of sin A:

1 / [(Opposite side)/(Hypotenuse)].

## Example

Cosecant is the reciprocal of sine.

And sine is SOH:

Sine, Opposite side, Hypotenuse.

But the side opposite to ∠A is unknown.

So set the opposite side x

and find x first.

The given triangle is a right triangle.

So, by the Pythagorean theorem,

x^{2} + 3^{2} = 4^{2}.

+3^{2} = +9

4^{2} = 16

Move +9 to the right side.

Then x^{2} = 7.

Square root both sides.

Then x = √7.

x is the opposite side.

So x is plus.

So you don't have to write ±.

Write √7

next to the opposite side.

csc A = 1/[sin A]

Find sin A.

Sine is SOH:

Sine,

Opposite side (√7),

Hypotenuse (4).

So 1/[sin A] = 1/[√7/4].

1/[√7/4] = 4/√7

The reciprocal of √7/4 is

4/√7.

Rationalize the denominator √7

by multiplying √7/√7.

Then 4√7/7.

So csc A = 4√7/7.