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, $180°=\mathrm{\pi } \mathrm{radians}$

so, $1°=\frac{\mathrm{\pi }}{180} radians$

Therefore, $k°=\frac{\mathrm{\pi }}{180}\times k radians$

 

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

Since, $\mathrm{\pi } \mathrm{radians} = 180°$

so, $1 \mathrm{radian} = {\left(\frac{180}{\mathrm{\pi }}\right)}^{°}$

Therefore, $k \mathrm{radian}s = {\left(\frac{180}{\mathrm{\pi }}\times k\right)}^{°}$

• 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 $k°=\frac{\mathrm{k\pi }}{180} radians$. Here $k=120$.

Therefore, $120°=\frac{120\pi }{180} radians$$=\frac{2\pi }{3} radians$

 

Example 2: Express $150°$ in radians.

Solution: We know that $k°=\frac{\mathrm{k\pi }}{180} radians$. Here $k=150$.

Therefore, $150°=\frac{150\pi }{180} radians$$=\frac{5\pi }{6} radians$

 

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

Solution: We know that $k°=\frac{\mathrm{k\pi }}{180} radians$. Here $k=-210$.

Therefore, $-210°=\frac{-210\pi }{180} radians$$=-\frac{7\pi }{6} radians$

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

 

Example 4: Convert $\frac{\mathrm{\pi }}{9} radians$ to degrees.

Solution: We know that $k radians={\left(\frac{180k}{\mathrm{\pi }}\right)}^{°}$. Here $k=\frac{\mathrm{\pi }}{9}$.

Therefore, $\frac{\mathrm{\pi }}{9} radians={\left(\frac{180\times {\displaystyle \frac{\mathrm{\pi }}{9}}}{\mathrm{\pi }}\right)}^{°}$$={\left(\frac{20\mathrm{\pi }}{\mathrm{\pi }}\right)}^{°}$$={20}^{°}$

 

Example 5: Express $-\frac{11\mathrm{\pi }}{20} radians$ to degrees.

Solution: We know that $k radians={\left(\frac{180k}{\mathrm{\pi }}\right)}^{°}$. Here $k=\frac{-11\mathrm{\pi }}{20}$.

Therefore, $\frac{-11\mathrm{\pi }}{20} radians={\left(\frac{180\times {\displaystyle \frac{-11\mathrm{\pi }}{20}}}{\mathrm{\pi }}\right)}^{°}$$={\left(\frac{-99\mathrm{\pi }}{\mathrm{\pi }}\right)}^{°}$$=-{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, $\sin \left(90°\right)=\sin \left(\frac{\mathrm{\pi }}{2}\right)=1$