# Caculatesin (θ)

θ angle

sin(θ)

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

## Formula

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

## Calculating Sin of an Angle

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

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

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

where:

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

### Example 1

Let's calculate the sine 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 sine of 30 degrees is:

\( \sin(30^{\circ}) = \frac{1}{2} = 0.5 \)

### 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 sine of 45 degrees is:

\( \sin(45^{\circ}) = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \approx 0.707 \)

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