# Caculatecsc (θ)

θ angle

csc(θ)

In a right angled triangle, Cosecant of an angle is the ratio of the length of the hypotenuse to the length of the opposite side.

## Formula

To find the Cosecant of a given angle θ , use the following formula.

## Calculating Cosecant of an Angle

The cosecant (csc) of an angle in a right triangle is a trigonometric function that represents the ratio of the length of the hypotenuse to the opposite side. It is the reciprocal of the sine function. This function is often used to find unknown side lengths or angles in right triangles.

The cosecant of an angle \( \theta \) is defined as:

\( \csc(\theta) = \frac{\text{hypotenuse}}{\text{opposite}} \)

where:

**hypotenuse**is the length of the hypotenuse (the side opposite the right angle).**opposite**is the length of the side opposite to angle \( \theta \).

### Example 1

Let's calculate the cosecant of a 30-degree angle:

In a right triangle, if the angle \( \theta = 30^{\circ} \), the opposite side is half the length of the hypotenuse. So, the cosecant of 30 degrees is:

\( \csc(30^{\circ}) = \frac{\text{hypotenuse}}{\text{opposite}} = \frac{1}{\sin(30^{\circ})} = \frac{1}{0.5} = 2 \)

### Example 2

Consider another example with a 45-degree angle:

In a right triangle, if the angle \( \theta = 45^{\circ} \), the opposite side and the adjacent side are of equal length, and the hypotenuse is \( \sqrt{2} \) times the length of the opposite side. So, the cosecant of 45 degrees is:

\( \csc(45^{\circ}) = \frac{\text{hypotenuse}}{\text{opposite}} = \frac{1}{\sin(45^{\circ})} = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2} \approx 1.414 \)

These examples demonstrate how to calculate the cosecant of an angle manually, showing the importance of understanding the trigonometric function cosecant in various mathematical and practical applications.