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# Positive and Negative Angles of Mathematics

Updated: Jun 25

An angle is a measure of rotation between two rays that share a common endpoint, called the vertex. Angles can be classified into positive and negative angles based on the direction of rotation from the initial side to the terminal side.

## What are Positive and Negative Angles?

A positive angle is formed by an anti-clockwise or counter-clockwise rotation from the initial side. For example, 30°, 90°, 180°, and 360° are all positive angles

A negative angle is formed by a clockwise rotation from the initial side. For example, -30°, -90°, -180°, and -360° are all negative angles.

The figure above shows some examples of positive and negative angles in standard position. An angle is in standard position if its vertex is at the origin and its initial side lies along the positive x-axis.

## How to Measure Angles?

There are two common ways to measure angles: degrees and radians. Degrees are based on dividing a full circle into 360 equal parts. Radians are based on the ratio of the arc length to the radius of a circle.

One degree is equal to 1/360 of a full circle. One radian is equal to the angle subtended by an arc whose length is equal to the radius of the circle.

The conversion between degrees and radians is given by the following formula:

## Why are Positive and Negative Angles Important?

Positive and negative angles are important for studying trigonometric functions, which relate angles to ratios of sides of right triangles. Trigonometric functions have many applications in geometry, physics, engineering, and other fields.

The basic trigonometric functions are sine, cosine, and tangent. They are defined as follows for an acute angle θ in a right triangle:

The values of these functions depend on the size and sign of the angle.

The sign of the function also depends on which quadrant the angle lies in. The following table summarizes the signs of the trigonometric functions in each quadrant:

The figure below shows the graphs of the sine, cosine, and tangent functions for angles from -360° to 360°. Notice how the functions repeat their values every 360°. This is called periodicity.

The trigonometric functions can also be extended to angles greater than 360° or less than -360° by adding or subtracting multiples of 360°. For example,

## Summary

In this blog, we learned about positive and negative angles of mathematics. We learned that

1. An angle is a measure of rotation between two rays that share a common endpoint, called the vertex.

2. A positive angle is formed by an anti-clockwise or counter-clockwise rotation from the initial side.

3. A negative angle is formed by a clockwise rotation from the initial side.

4. Angles can be measured in degrees or radians. Degrees are based on dividing a full circle into 360 equal parts. Radians are based on the ratio of the arc length to the radius of a circle.

5. Positive and negative angles are important for studying trigonometric functions, which relate angles to ratios of sides of right triangles. Trigonometric functions have many applications in geometry, physics, engineering, and other fields.