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 360°2=180°. Similarly, a quadrant subtends 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 2π radians.
  • Therefore, we can conclude that 180°=π radians.

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

Since, 180°=π radians

so, 1°=π180 radians

Therefore, k°=π180×k radians

 

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

Since, π radians = 180°

so, 1 radian = 180π°

Therefore, k radians = 180π×k°

  • 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 π6 π4 π3 π2 π 3π2 2π

 

Solved Examples

Example 1: Convert 120° into radians.

Solution: We know that k°=180 radians. Here k=120.

Therefore, 120°=120π180 radians=2π3 radians

 

Example 2: Express 150° in radians.

Solution: We know that k°=180 radians. Here k=150.

Therefore, 150°=150π180 radians=5π6 radians

 

Example 3: Convert -210° to radians.

Solution: We know that k°=180 radians. Here k=-210.

Therefore, -210°=-210π180 radians=-7π6 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 π9 radians to degrees.

Solution: We know that k radians=180kπ°. Here k=π9.

Therefore, π9 radians=180×π9π°=20ππ°=20°

 

Example 5: Express -11π20 radians to degrees.

Solution: We know that k radians=180kπ°. Here k=-11π20.

Therefore, -11π20 radians=180×-11π20π°=-99ππ°=-99°

Cheat Sheet

  • To convert the measure of an angle from degrees to radians, multiply the given angle byπ180 to get the result in radians.
  • Likewise, to convert the measure of an angle from radians to degrees, multiply the given angle by 180π 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 90°=sin π2=1