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,
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 denominator √7
by multiplying √7/√7.
Then 4√7/7.
So csc A = 4√7/7.