Cosecant: in a Right Triangle

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

Formula

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

Example

Solution

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,
x2 + 32 = 42.

+32 = +9

42 = 16

Move +9 to the right side.
Then x2 = 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 denominator7
by multiplying √7/√7.

Then 4√7/7.

So csc A = 4√7/7.