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# Precalculus - Degree & Radians

## Introduction

• In geometry, we have seen that when two rays originate from a common point (called a vertex), an angle is formed between them.
• The measure of an angle indicates the amount of rotation between the two rays forming the angle.
• Two commonly used systems to measure angles are degrees and radians.
• Angles measured in 'degrees' are denoted by a small circle as a superscript such as $30°$$90°$, etc.
• If we divide the circumference of a circle (of any radius) into 360 equal arcs, then the measure of the angle subtended at the center by one such arc is termed as one degree.
• Thus, the measure of one complete revolution (circumference of a circle) is $360°$, and hence, we can determine the measure of any angle if we know the proportion it represents of an entire revolution (or circumference of a circle).
• For example, a semi-circle subtends $\frac{360°}{2}=180°$. Similarly, a quadrant subtends $\frac{360°}{4}=90°$.
• One radian is the measure of an angle subtended at the center of a circle by an arc whose length is equal to the radius of that circle.
• In general, the angle measured in the counter-clockwise sense is taken as positive, while the angle measured in a clockwise direction is taken as negative.

## Relationship between degrees and radians

• We have seen that one revolution (or circumference of a circle) subtends $360°$, and the same is equivalent to $2\mathrm{\pi }$ radians.
• Therefore, we can conclude that $180°=\mathrm{\pi }$ radians.

1. Converting the measure of an angle from degrees to radians

Since,

so,

Therefore,

2. Converting the measure of an angle from radians to degrees

Since,

so,

Therefore,

• Some of the degree-radian equivalent values worth remembering are given in the table below.
 Angle in degrees $0°$ $30°$ $45°$ $60°$ $90°$ $180°$ $270°$ $360°$ Angle in radians $0$ $\frac{\mathrm{\pi }}{6}$ $\frac{\mathrm{\pi }}{4}$ $\frac{\mathrm{\pi }}{3}$ $\frac{\mathrm{\pi }}{2}$ $\mathrm{\pi }$ $\frac{3\mathrm{\pi }}{2}$ $2\mathrm{\pi }$

## Solved Examples

Example 1: Convert $120°$ into radians.

Solution: We know that . Here $k=120$.

Therefore,

Example 2: Express $150°$ in radians.

Solution: We know that . Here $k=150$.

Therefore,

Example 3: Convert $-210°$ to radians.

Solution: We know that . Here $k=-210$.

Therefore,

Note: Negative sign here indicates that the angle is measured in the clockwise direction with respect to the positive x-axis.

Example 4: Convert  to degrees.

Solution: We know that . Here $k=\frac{\mathrm{\pi }}{9}$.

Therefore, $={\left(\frac{20\mathrm{\pi }}{\mathrm{\pi }}\right)}^{°}$$={20}^{°}$

Example 5: Express  to degrees.

Solution: We know that . Here $k=\frac{-11\mathrm{\pi }}{20}$.

Therefore, $={\left(\frac{-99\mathrm{\pi }}{\mathrm{\pi }}\right)}^{°}$$=-{99}^{°}$

## Cheat Sheet

• To convert the measure of an angle from degrees to radians, multiply the given angle by$\frac{\mathrm{\pi }}{180}$ to get the result in radians.
• Likewise, to convert the measure of an angle from radians to degrees, multiply the given angle by $\frac{180}{\mathrm{\pi }}$ to get the result in degrees.

## Blunder Areas

• The numerical value of any trigonometric ratio at an angle remains the same irrespective of the system chosen for its representation. For example,