For The Angle of 120 Degrees Calculate Sin 120

What is sin(120°)?

The sine of an angle in a right triangle is defined as the ratio of the length of the opposite side to the hypotenuse. For 120 degrees, which is in the second quadrant of the unit circle, the sine value is positive.

In trigonometry, the sine function is periodic with a period of 360°, meaning sin(θ) = sin(θ + 360°n) for any integer n. This property allows us to find sin(120°) by relating it to other angles we know.

How to calculate sin(120°)

There are several methods to calculate sin(120°):

1. Using reference angles
2. Using the unit circle
3. Using trigonometric identities
4. Using a calculator

Method 1: Using reference angles

120° is located in the second quadrant. The reference angle for 120° is calculated as 180° - 120° = 60°.

In the second quadrant, sine is positive, so sin(120°) = sin(60°).

We know that sin(60°) = √3/2 ≈ 0.8660.

Method 2: Using the unit circle

On the unit circle, the y-coordinate of the point at 120° gives sin(120°).

Coordinates for 120°: (cos(120°), sin(120°)) = (-1/2, √3/2).

Therefore, sin(120°) = √3/2 ≈ 0.8660.

Method 3: Using trigonometric identities

We can use the angle addition formula:

Let's express 120° as 180° - 60°:

We know:

• sin(180°) = 0
• cos(180°) = -1
• cos(60°) = 1/2
• sin(60°) = √3/2

Substituting these values:

sin(120°) reference angle

The reference angle for 120° is 60° because 120° is 60° away from the x-axis in the second quadrant.

Since sine is positive in the second quadrant, we have:

This reference angle method provides a quick way to find the sine of 120° without needing to draw the angle.

sin(120°) on the unit circle

The unit circle is a circle with radius 1 centered at the origin (0,0) in the coordinate plane. Any angle θ corresponds to a point (cosθ, sinθ) on the unit circle.

For 120°:

• cos(120°) = -1/2
• sin(120°) = √3/2

This means the point on the unit circle at 120° is (-0.5, 0.8660). The y-coordinate of this point is sin(120°).

Visualizing the unit circle helps understand the relationship between angles and their trigonometric values.

sin(120°) calculator

Our online calculator provides an easy way to find sin(120°) and other trigonometric values. Simply enter the angle in degrees and click "Calculate".

The calculator uses precise mathematical algorithms to compute trigonometric functions with high accuracy.

For 120°:

This matches our manual calculations, confirming the accuracy of our calculator.