The cosecant of an angle is defined with respect to the sides of a right triangle. In a right triangle, the cosecant is equal to the length of the hypotenuse divided by the side opposite the angle. The cosecant is the reciprocal function of the sine.

Here, we will learn more about the cosecant and use diagrams to facilitate understanding. Also, we will obtain the values of the cosecant of the most important angles. Finally, we will see some practice exercises.

## Definition of cosecant of an angle

The cosecant of an angle is defined using a right triangle. The cosecant of an angle is equal to the length of the hypotenuse divided by the length of the side opposite the angle in the triangle.

Another definition of cosecants is that they are the reciprocal functions of the sine. This means that the cosecant of an angle is equal to 1 divided by the sine of the angle. We know that the sine is equal to the opposite side divided by the hypotenuse and the cosecant is the inverse of that. Therefore, we have:

$latex \csc (\theta)=\frac{1}{\sin}=\frac{H}{O}$

where H is the hypotenuse and O is the opposite side.

## Formula for the cosecant in right triangles

We can find the formula for the cosecant of the angles in the following right triangle ABC.

Generally, we use capital letters to represent the angles and lowercase letters to represent the sides of the triangle opposite the corresponding angle. For example, side *a* is opposite angle A, side *b* is opposite angle B, and side *c* is opposite angle C.

We know that the cosecant of an angle in a right triangle is equal to the hypotenuse divided by the opposite side:

$latex \csc=\frac{\text{hypotenuse}}{\text{opposite}}$ |

In the triangle above, we have $latex \csc(A)= \frac{c}{a}$ and also $latex \csc(B)= \frac{c}{b}$.

## Cosecant for common special angles

In the triangle above, we have $latex \csc(A) = \frac{c}{a}$ and also $latex \csc(B) = \frac{c}{b}$. The cosecant values for common angles can be obtained using the proportions of the sides of special triangles.

For example, we can find the value of the cosecant of 45° using the 45°-45°-90° triangle. The proportions of this triangle are found using the Pythagorean theorem: $latex {{c}^2}={{a}^2}+{{b}^2}$.

Since we have two equal 45° angles, we know that $latex a = b$, so the Pythagorean theorem becomes $latex {{c}^2}=2{{a}^2}$. This means that $latex c=a \sqrt{2}$. Therefore, the cosecant of 45° is equal to $latex \sqrt{2}$.

The cosecant values of 30° and 60° are found using the 30°-60°-90° triangle. This triangle has the proportions 1:$latex \sqrt{3}$: 2. Using this, we have $latex \csc(30^{\circ})=2$ and $latex \csc(60^{\circ}) = \frac{2}{\sqrt{3}}$, which is equivalent to $latex \frac{ 2 \sqrt{3}}{3}$.

Furthermore, we can find the values of the cosecants of the angles 0° and 90° using the unit circle. When the angle is 0°, the opposite side equals 0 and we cannot divide by 0, so the cosecant is undefined.

When the angle is 90°, the opposite side is equal to 1 and the hypotenuse is equal to 1, so the cosecant is equal to 1.

Degrees | Radians | Cosecant |

90° | $latex \frac{\pi}{2}$ | 1 |

60° | $latex \frac{\pi}{3}$ | $latex \frac{2\sqrt{3}}{3}$ |

45° | $latex \frac{\pi}{4}$ | $latex \sqrt{2}$ |

30° | $latex \frac{\pi}{6}$ | $latex 2$ |

0° | 0 | Undefined |

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## Cosecant of an angle – Examples with answers

The relationship between the cosecant and the sides of a right triangle is used to solve the following examples. Each of the following examples refers to the right triangle seen above, so the notation of the sides is the same.

**EXAMPLE 1**

If we have $latex \csc(A)=1.4$ and $latex a = 6$, what is the value of *c*?

##### Solution

Referring to the right triangle above, we have $latex \csc(A) = \frac{c}{a}$. We use this relationship along with the given values to find the value of *c*:

$latex \csc(A)=\frac{c}{a}$

$latex 1.4=\frac{c}{6}$

$latex c=1.4(6)$

$latex c=8.4$

he value of the *c* side is equal to 8.4.

**EXAMPLE 2**

Determine the value of *b* if we have $latex c = 12$ and $latex \csc(B)=2.9$.

##### Solution

We have the values $latex c=12$ and $latex \csc (B)=2.9$. Using these values in the relation $latex \csc(B) = \frac{c}{b}$, we have:

$latex \csc(B)=\frac{c}{b}$

$latex 2.9=\frac{12}{b}$

$latex b=\frac{12}{2.9}$

$latex b=4.14$

The value of *b* is 4.14.

**EXAMPLE 3**

We have the values $latex c=4$ and $latex a = 2 \sqrt{3}$. What is the value of angle A?

##### Solution

We use the right triangle above as a reference to form the relation $latex \csc(A) = \frac{c}{a}$. Therefore, using the given values, we have:

$latex \csc(A)=\frac{c}{a}$

$latex \csc(A)=\frac{4}{2\sqrt{3}}$

$latex \csc(A)=\frac{2\sqrt{3}}{3}$

We can use a calculator with $latex {{\csc}^{- 1}}$ or the table above to determine that:

$latex A=60$°

Angle A measures 60°.

→ Cosecant Calculator (Degrees and Radians)

## Cosecant of an angle – Practice problems

Solve the following practice problems using what you have learned about the cosecant of an angle. These problems refer to the right triangle seen above, so they use the same notation for sides and angles.

## See also

Interested in learning more about secant, cosecant, and cotangent? Take a look at these pages:

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