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 1 answerLast reply by: Dr. William MurrayThu Apr 25, 2013 2:10 PMPost by Erin Murphy on March 16, 2010You are a fantastic prof. On to my next lesson! 1 answerLast reply by: Dr. William MurrayThu Apr 25, 2013 2:12 PMPost by Sheila Greenfield on March 3, 2011i get this and i'm a freshman in high school i really like this cant wait to learn more 1 answerLast reply by: Dr. William MurrayThu Apr 25, 2013 2:15 PMPost by David Burns on August 8, 2011I wish this site had tests available, or at least links to them. Other than that I love it here. 1 answerLast reply by: Dr. William MurrayThu Apr 25, 2013 2:16 PMPost by Robert Reynolds on October 22, 2011Thumbs up for 2 things:1. Assessment test at the beginning to find where you are at now.2. End of lesson tests.That make this site the Deathstar of education. (Without the silly hole that you can shoot down and blow the whole thing.) 1 answerLast reply by: Dr. William MurrayThu Apr 25, 2013 2:18 PMPost by Kyle Spicer on December 6, 2011where do you take the assessment test? I can't find it. 1 answerLast reply by: Dr. William MurrayThu Apr 25, 2013 2:50 PMPost by Valtio Cooper on January 14, 2012Great lecture! I got it but I'm having a problem with a question that I got for homework pertaining to this topic! I was wondering if i could be given some guidelines if possible please. The question is:A Hexagon is inscribed in a circle. if the difference between the area of the circle and the area of the hexagon is 24meters squared use the formula for the area of the sector to approximate the radius of the circle. 1 answerLast reply by: Dr. William MurrayThu Apr 25, 2013 2:53 PMPost by Janet Wyatt on February 10, 2012Is there practice worksheets I can print? 1 answerLast reply by: Dr. William MurrayThu Apr 25, 2013 2:59 PMPost by kirill frusin on March 2, 2012I believe you confused compliment and supplement in one of your videos. The video I watched before this about RADIANS says supplement is two angles added to be 90 degrees and complimentary add to 180 degrees. 1 answerLast reply by: Dr. William MurrayThu Apr 25, 2013 3:02 PMPost by Levi Stafford on March 19, 2012commenting on the text on the quick notes. "2π parts, denoted 2π R." the pi's look like "n's" and it is confusing...I thought they were variables. 1 answerLast reply by: Dr. William MurrayThu Apr 25, 2013 3:04 PMPost by Edmund Mercado on April 15, 2012A very fine presentation.

### 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:

1. Convert 27° into radians.
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]

## Convert 185° into radians

• Recall the equation for converting degrees to radians
• degree measure ×[(π)/(180°)] = radian measure
• 185° ×[(π)/(180°)] =
• [(185°π)/(180°)]
• The degrees will cancel out, now simplify your answer

## 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.

## Angles

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.