### Angles

Main definitions and formulas:

• Degrees are a unit of measurement by which a circle is divided into 360 equal parts, denoted 360° .
• Radians are a unit of measurement by which a circle is divided into 2π parts, denoted 2π R.
• Since the circumference of a circle is 2π r, this means that a 1R angle cuts off an arc whose length is exactly equal to the radius. (It is [1/(2π )] of the whole circle.)
• Since 2π ≈ 6.28..., this means that 1R is about one sixth of a circle. But we seldom use whole numbers of radians. Instead we use multiples of π . For example, (π /2)R is exactly one fourth of a circle.
•  degree measure × π 180 = radian measure
•  radian measure × 180 π = degree measure
• Coterminal angles are angles that differ from each other by adding or subtracting multiples of 2π R (i.e. 360° ). If you graph them in the coordinate plane starting at the x-axis, they terminate at the same place.
• Complementary angles add to (π /2)R(i.e. 90° ).
• Supplementary angles add to π R (i.e. 180° ).

Example 1:

If a circle is divided into 18 equal angles, how big is each one, in degrees and radians?

Example 2:

2. Convert (5π /12)R into degrees.

Example 3:

For each of the following angles, determine which quadrant it is in and find a coterminal angle between 0 and 360° or between 0 and 2π R.
1. 1000°
2. − (19π /6)R
3. -586°
4. (22π /7)R

Example 4:

Convert the following "common values" from degrees to radians: 0, 30, 45, 60, 90. Find the complementary and supplementary angles for each one, in both degrees and radians.

Example 5:

For each of the following angles, determine which quadrant it is in and find a coterminal angle between 0 and 360° or between 0 and 2π R.
1. − (5π /4)R
2. 735°
3. − (7π /3)R
4. -510°

## A circle is divided into 12 equal angles. Calculate the measure of each angle in degrees and radians.

• In order to calculate the degrees and radians, recall that a circle is 360°
• Calculate the measure of each angle in degrees first
• [360/12]° = 30°
• In radians, we know that 360° is 2π. So, we can calculate the measure of each angle in radians
• [(2π)/12] = [(π)/6]

### 30° and [(π)/6]

• Recall the equation for converting degrees to radians
• degree measure ×[(π)/(180°)] = radian measure
• 185° ×[(π)/(180°)] =
• [(185°π)/(180°)]

## Convert [(7π)/10] into degrees

• Recall the equation for converting radians to degrees
• radian measure ×[(180°)/(π)] = degree measure
• [(7π)/10] ×[(180° )/(π)] =
• 7 ×18°

## What is the degree measure of an arc whose measure is [(3π)/5] radians?

• radian measure ×[(180°)/(π)]= degree measure
• [(3π)/5] ×[(180°)/(π)] =
• [(540°π)/(5π)]
• The radians cancel so now just divide

## What is the radian measure of an arc whose measure is 76 °?

• Recall the formula for converting degrees to radians
• degree measure ×[(π)/(180°)] = radian measure
• 76° × [(π)/(180° )] =
• [(76° π)/(180° )]
• The degrees will cancel. Simplify your fraction

## Determine which quadrant the following angle is in and find a coterminal angle between 0° and 360° : 450°

• Notice that 450° is larger than 360°, so we must subtract 360° from our given angle until we reach an angle that is between 0° and 360°
• 450° - 360° = 115° which is an angle that is between 0° and 360° and it is coterminal to 450°
• Now we can determine which quadrant our angle is in by using the following:
Quadrant I has angles between 0° and 90°
Quadrant II has angles between 90° and 180°
Quadrant III has angles between 180° and 270°
Quadrant IV has angles between 270° and 360°

## Determine which quadrant the following angle is in and find a coterminal angle between 0° and 360° : [( − 13π)/5]

• Notice that [( − 13π)/15] is smaller than 0°, so we must add 360° or 2π to our given angle until we reach an angle that is between 0 and 360° (i.e. between 0 and 2π)
• [( − 13π)/15] + 2π = [(17π)/15] which is an angle that is between 0 and 2π and it is coterminal to [( − 13π)/15]
• Now we can determine which quadrant our angle is in by using the following:
Quadrant I has angles between 0 and [(π)/2]
Quadrant II has angles between [(π)/2] and π
Quadrant III has angles between π and [(3π)/2]
Quadrant IV has angles between [(3π)/2]and 2π

## Find the complementary angle for each of the following angles: a. 47° b. [(π)/12]

• Recall that complementary angles add to 90° or [(π)/2]

## Find the supplementary angle for each of the following angles: a. 114° b. [(4π)/5]

• Recall that supplementary angles add to 180° or π
• a. 180° − 114° = 66°
b. π− [(4π)/5] = [(π)/5]

## For each of the following angles, determine which quadrant it is in and find a coterminal angle between 0° and 360° or between 0 and 2π a. [( − 6π)/11] b. 623° c. [( − 27π)/13] d. 1572°

• a. [( − 6π)/11]is smaller than 0 so you have to add 2π to find a coterminal angle
• [( − 6π)/11] + 2π = [(16π)/11] which is in quadrant III
• b. 623° is larger than 360° so you have to subtract 360° to find a coterminal angle
• 623° − 360° = 263° which is in quadrant III
• c. [( − 27π)/13] is smaller than 0 so you have to add 2p to find a coterminal angle
• [( − 27π)/13] + 2π = [( − π)/13] which is still smaller than 0 so keep adding 2p
• [( − π)/13] + 2π = [(25π)/13] which is in quadrant IV
• d. 1572° is larger than 360° so you have to subtract 360° to find the coterminal angle
• 1572° − 360° = 1212° which is still larger than 360° so keep subtracting by 360°
• 1212° − 360° = 852°
852° − 360° = 492°
492° − 360° = 132° which is in quadrant II

### a. [(16π)/11]; III b. 263°; III c. [(25π)/13]; IV d. 132°; II

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.