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8.03 Degrees, radians, and angles as rotations


Angles as rotations

Previously, we have defined angles as geometric objects - two noncollinear rays that share a common endpoint, or vertex. We can also define an angle as the action of rotating a ray about its endpoint.

Using this rotational definition, we define the starting position of the ray as the initial side of the angle. The ray's position after the rotation forms the terminal side of the angle.

If we view the angle in the coordinate plane, we say that the angle is in standard position if its vertex is at the origin and its initial side is along the positive $x$x-axis.

An angle in standard position


By defining an angle as a rotation, we can also allow for the concept of positive and negative angles. A positive angle denotes a counterclockwise rotation in standard position and a negative angle denotes a clockwise rotation.

A counterclockwise rotation create a positive angle

A clockwise rotation creates a negative angle


Measuring angles

From the time of the ancient Babylonians, it has been the practice to divide circles into $360$360 small arcs. The angle subtended at the center by any one of those arcs, is called one degree. In effect, an arc of the circle is used as a measure of the angle it subtends.

In a similar way, we now restrict our attention to circles of radius one unit, this is called the unit circle. We measure angles subtended at the center by arcs of this circle. This method of measuring angles is called radian measure.

Do you remember how to find the circumference of a circle?  We use the formula $C=2\pi r$C=2πr.  So if the radius ($r$r) is $1$1, then the circumference is $2\pi$2π.

The angle represented by a full turn around the circle is $2\pi$2π radians. This is equivalent to $360^\circ$360°

A half-circle makes an angle of $\pi$π radians or $180^\circ$180° 

and a right-angle is $\frac{\pi}{2}$π2 radians. 

An angle of $1$1 radian must be $\frac{360^\circ}{2\pi}\approx57.3^\circ$360°2π57.3° .

In practice, angles given in radian measure are usually expressed as fractions of $\pi$π

Because angles in radian measure are in essence just fractions of the unit circle, they do not require a unit.

Converting between degrees and radians

Since $360^\circ=2\pi$360°=2π radians, it follows that $180^\circ=\pi$180°=π radians and $1^\circ=\frac{\pi}{180}$1°=π180 radians.

Worked example

Question 1

Complete the table below to find the corresponding radian measures of each angle.

Fraction of a circle $1$1 (Full Circle) $\frac{1}{2}$12 $\frac{1}{3}$13 $\frac{1}{4}$14 $\frac{1}{6}$16 $\frac{1}{8}$18 $\frac{1}{12}$112 $\frac{1}{24}$124 $\frac{1}{36}$136 $\frac{1}{360}$1360
Measure in degrees $360^\circ$360° $180^\circ$180° $120^\circ$120° $90^\circ$90° $60^\circ$60° $45^\circ$45° $30^\circ$30° $15^\circ$15° $10^\circ$10° $1^\circ$1°
Measure in radians $2\pi$2π $\pi$π                

Think: Since a full circle is $360^\circ$360°, we know that $120^\circ$120° is $\frac{1}{3}$13 of a circle. Since we divide $360$360 by $3$3 to get $120$120, we should also divide $2\pi$2π by $3$3  to find the number of radians. Therefore, $120^\circ=\frac{2\pi}{3}$120°=2π3 radians.

Do: Complete the rest of the table in a similar way.

Fraction of a circle $1$1 (Full Circle) $\frac{1}{2}$12 $\frac{1}{3}$13 $\frac{1}{4}$14 $\frac{1}{6}$16 $\frac{1}{8}$18 $\frac{1}{12}$112 $\frac{1}{24}$124 $\frac{1}{36}$136 $\frac{1}{360}$1360
Measure in degrees $360^\circ$360° $180^\circ$180° $120^\circ$120° $90^\circ$90° $60^\circ$60° $45^\circ$45° $30^\circ$30° $15^\circ$15° $10^\circ$10° $1^\circ$1°
Measure in radians $2\pi$2π $\pi$π $\frac{2\pi}{3}$2π3 $\frac{2\pi}{4}$2π4 $\frac{2\pi}{6}$2π6 $\frac{2\pi}{8}$2π8 $\frac{2\pi}{12}$2π12 $\frac{2\pi}{24}$2π24 $\frac{2\pi}{36}$2π36 $\frac{2\pi}{360}$2π360

Simplify the fractions, if possible

Fraction of a circle $1$1 (Full Circle) $\frac{1}{2}$12 $\frac{1}{3}$13 $\frac{1}{4}$14 $\frac{1}{6}$16 $\frac{1}{8}$18 $\frac{1}{12}$112 $\frac{1}{24}$124 $\frac{1}{36}$136 $\frac{1}{360}$1360
Measure in degrees $360^\circ$360° $180^\circ$180° $120^\circ$120° $90^\circ$90° $60^\circ$60° $45^\circ$45° $30^\circ$30° $15^\circ$15° $10^\circ$10° $1^\circ$1°
Measure in radians $2\pi$2π $\pi$π $\frac{2\pi}{3}$2π3 $\frac{\pi}{2}$π2 $\frac{\pi}{3}$π3 $\frac{\pi}{4}$π4 $\frac{\pi}{6}$π6 $\frac{\pi}{12}$π12 $\frac{\pi}{18}$π18 $\frac{\pi}{180}$π180

Reflect: What patterns exist in the table of values above?


Practice questions


Convert $90^\circ$90° to radians.

Give your answer in exact form.


Convert $-300^\circ$300° to radians.

Give your answer in exact form.


Convert $\frac{2\pi}{3}$2π3 radians to degrees.


Convert $4.2$4.2 radians to degrees.

Give your answer correct to one decimal place.


Coterminal angles

Using the rotation definition of an angle, it's also possible to have an angle that rotates more than once around the circle. Rotations of this type will have measures with a magnitude greater than $360^\circ$360°(or $2\pi$2π).

Because of the rotation definition of an angle, it's possible to have two angles with the same initial and terminal sides but different measures. Angles that are related in this way are called coterminal angles.

Coterminal angles have measures that differ by an integer multiple of $2\pi$2π.

In general, an angle coterminal with another angle differs from it by an integer multiple of $2\pi$2π (if measured in radians) or $360^{\circ}$360 (if measured in degrees).  


Worked examples

Question 6

List: Two negative and two positive angles that are coterminal with $115^{\circ}$115.

Think: We need only add or subtract multiples of $360^{\circ}$360 to obtain the coterminal angles.




Question 7

Find: The coterminal angle closest to zero for $\frac{39\pi}{4}$39π4.

Think: The number $\frac{39\pi}{4}$39π4 can be written as $\frac{36\pi+3\pi}{4}=9\pi+\frac{3\pi}{4}$36π+3π4=9π+3π4, so it is between $9\pi$9π and $10\pi$10π.

Do: We can subtracting $5\times2\pi$5×2π. This gives $\frac{39\pi}{4}-10\pi=-\frac{\pi}{4}$39π410π=π4. The closest coterminal angle to zero is therefore $\frac{-\pi}{4}$π4.

Question 8

Where: In which quadrant does the angle $7440^{\circ}$7440 lie?

Think: The strategy will be to remove integer multiples of $360^{\circ}$360 until an angle between $0^{\circ}$0 and $360^{\circ}$360 is reached.

Do: By division, we see that $20\times360<7440<21\times360$20×360<7440<21×360.

$7440^\circ-20\times360^\circ$7440°20×360° $=$= $7440^\circ-7200^\circ$7440°7200°
  $=$= $240^\circ$240°

This angle is greater than $180^{\circ}$180 and less than $270^{\circ}$270 and is therefore in the fourth quadrant.


Practice Questions

Question 9

Find the angle of smallest positive measure that is coterminal with a $489^\circ$489° angle.

Question 10

Consider an angle of $-58$58°.

  1. Find the angle of smallest positive measure that is coterminal with $-58$58°.

  2. Find the angle of smallest negative measure that is coterminal with $-58$58°.

  3. Which quadrant does $\left(-58\right)^\circ$(58)° lie in?

    quadrant $3$3


    quadrant $1$1


    quadrant $4$4


    quadrant $2$2


Question 11

State the expression in terms of $n$n, where $n$n represents any integer, that generates all angles coterminal with $\frac{\pi}{2}$π2.

  1. $\frac{\pi}{2}$π2$+$+$\editable{}$


Reference angles

In the circle with radius $1$1 centered at the origin, we measure angles of any magnitude between the positive $x$x-axis and the radius to a point that moves on the circle. The trigonometric functions of those angles are defined in a manner that guarantees that a function of any angle will be related to the same function of an angle in the first quadrant. We often use $\theta$θ for the angle and $\alpha$α for the acute reference angle.

The angle $\theta$θ measuring $265^\circ$265° has a reference angle $\alpha$α of $85^\circ$85°

The first quadrant or acute angle to which a particular angle is related in this way may be called a reference angle. The reference angle is between $0^\circ$0° and $90^\circ$90°.

To find a reference angle, first, if necessary, add or subtract multiples of $360^\circ$360°  to obtain an angle between $0^\circ$0° and $360^\circ$360°. Then, decide what quadrant the angle is in.

If the resulting angle is in the first quadrant, it is the reference angle. $\alpha=\theta$α=θ

If it is in the second quadrant, subtract it from $180^\circ$180° to obtain the reference angle. $\alpha=180^\circ-\theta$α=180°θ

If it is in the third quadrant, subtract $180^\circ$180° from the angle. $\alpha=\theta-180^\circ$α=θ180°

In the fourth quadrant, subtract the angle from $360^\circ$360°. $\alpha=360^\circ-\theta$α=360°θ


Worked example

Question 12

Find: the acute reference angle for the angle $-534^\circ$534°

Think: First, we need to find the angle with the same terminal side which is between $0^\circ$0° and $360^\circ$360°.

Do: We add $360^\circ$360° twice to obtain $186^\circ$186°, an angle in the range $0^\circ$0° to $360^\circ$360°

The angle $186^\circ$186° is in the third quadrant. So, we subtract $180^\circ$180° from it. Hence, the reference angle is $6^\circ$6°.


Practice questions

question 13

Find the reference angle for $197^\circ$197°.

question 14

Point $P$P on the unit circle shows a rotation of $330^\circ$330°.

What acute angle in the first quadrant can $330^\circ$330° be related to?

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question 15

We want to find the reference angle for $519^\circ$519°.

  1. First find the coterminal angle to $519^\circ$519° that is between $0^\circ$0° and $360^\circ$360°.

  2. Hence find the reference angle for $519^\circ$519°.




Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

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