# Trigonometry Angles

Section 1: Trigonometric Functions: Lecture 1 | 39:05 min

There are two ways to measure angles: degrees and radians. When measuring in the degree unit, a circle is 360 degrees, a semicircle is 180 degrees, and a right angle is 90 degrees. And when measuring in the radians unit, a circle is 2π radians, half a circle is π radians and a right angle is π/2 radians. Fortunately, there are formulas for converting between the degrees and the radians, and you'll have the opportunity to practice those conversions and learn more about them through the examples. Additionally, in this lecture, you'll learn about complementary and supplementary angles, and what coterminal angles and quadrants are.

Table of Contents

39m 5s

- Intro0:00
- Degrees0:22
- Circle is 360 Degrees0:48
- Splitting a Circle1:13
- Radians2:08
- Circle is 2 Pi Radians2:31
- One Radian2:52
- Half-Circle and Right Angle4:00
- Converting Between Degrees and Radians6:24
- Formulas for Degrees and Radians6:52
- Coterminal, Complementary, Supplementary Angles7:23
- Coterminal Angles7:30
- Complementary Angles9:40
- Supplementary Angles10:08
- Example 1: Dividing a Circle10:38
- Example 2: Converting Between Degrees and Radians11:56
- Example 3: Quadrants and Coterminal Angles14:18
- Extra Example 1: Common Angle Conversions-1
- Extra Example 2: Quadrants and Coterminal Angles-2

43m 16s

- Intro0:00
- Sine and Cosine0:15
- Unit Circle0:22
- Coordinates on Unit Circle1:03
- Right Triangles1:52
- Adjacent, Opposite, Hypotenuse2:25
- Master Right Triangle Formula: SOHCAHTOA2:48
- Odd Functions, Even Functions4:40
- Example: Odd Function4:56
- Example: Even Function7:30
- Example 1: Sine and Cosine10:27
- Example 2: Graphing Sine and Cosine Functions14:39
- Example 3: Right Triangle21:40
- Example 4: Odd, Even, or Neither26:01
- Extra Example 1: Right Triangle-1
- Extra Example 2: Graphing Sine and Cosine Functions-2

33m 5s

- Intro0:00
- 45-45-90 Triangle and 30-60-90 Triangle0:08
- 45-45-90 Triangle0:21
- 30-60-90 Triangle2:06
- Mnemonic: All Students Take Calculus (ASTC)5:21
- Using the Unit Circle5:59
- New Angles6:21
- Other Quadrants9:43
- Mnemonic: All Students Take Calculus10:13
- Example 1: Convert, Quadrant, Sine/Cosine13:11
- Example 2: Convert, Quadrant, Sine/Cosine16:48
- Example 3: All Angles and Quadrants20:21
- Extra Example 1: Convert, Quadrant, Sine/Cosine-1
- Extra Example 2: All Angles and Quadrants-2

52m 3s

- Intro0:00
- Amplitude and Period of a Sine Wave0:38
- Sine Wave Graph0:58
- Amplitude: Distance from Middle to Peak1:18
- Peak: Distance from Peak to Peak2:41
- Phase Shift and Vertical Shift4:13
- Phase Shift: Distance Shifted Horizontally4:16
- Vertical Shift: Distance Shifted Vertically6:48
- Example 1: Amplitude/Period/Phase and Vertical Shift8:04
- Example 2: Amplitude/Period/Phase and Vertical Shift17:39
- Example 3: Find Sine Wave Given Attributes25:23
- Extra Example 1: Amplitude/Period/Phase and Vertical Shift-1
- Extra Example 2: Find Cosine Wave Given Attributes-2

36m 4s

- Intro0:00
- Tangent and Cotangent Definitions0:21
- Tangent Definition0:25
- Cotangent Definition0:47
- Master Formula: SOHCAHTOA1:01
- Mnemonic1:16
- Tangent and Cotangent Values2:29
- Remember Common Values of Sine and Cosine2:46
- 90 Degrees Undefined4:36
- Slope and Menmonic: ASTC5:47
- Uses of Tangent5:54
- Example: Tangent of Angle is Slope6:09
- Sign of Tangent in Quadrants7:49
- Example 1: Graph Tangent and Cotangent Functions10:42
- Example 2: Tangent and Cotangent of Angles16:09
- Example 3: Odd, Even, or Neither18:56
- Extra Example 1: Tangent and Cotangent of Angles-1
- Extra Example 2: Tangent and Cotangent of Angles-2

27m 18s

- Intro0:00
- Secant and Cosecant Definitions0:17
- Secant Definition0:18
- Cosecant Definition0:33
- Example 1: Graph Secant Function0:48
- Example 2: Values of Secant and Cosecant6:49
- Example 3: Odd, Even, or Neither12:49
- Extra Example 1: Graph of Cosecant Function-1
- Extra Example 2: Values of Secant and Cosecant-2

32m 58s

- Intro0:00
- Arcsine Function0:24
- Restrictions between -1 and 10:43
- Arcsine Notation1:26
- Arccosine Function3:07
- Restrictions between -1 and 13:36
- Cosine Notation3:53
- Arctangent Function4:30
- Between -Pi/2 and Pi/24:44
- Tangent Notation5:02
- Example 1: Domain/Range/Graph of Arcsine5:45
- Example 2: Arcsin/Arccos/Arctan Values10:46
- Example 3: Domain/Range/Graph of Arctangent17:14
- Extra Example 1: Domain/Range/Graph of Arccosine-1
- Extra Example 2: Arcsin/Arccos/Arctan Values-2

31m 8s

- Intro0:00
- Inverse Trigonometric Function Domains and Ranges0:31
- Arcsine0:41
- Arccosine1:14
- Arctangent1:41
- Example 1: Arcsines of Common Values2:44
- Example 2: Odd, Even, or Neither5:57
- Example 3: Arccosines of Common Values12:24
- Extra Example 1: Arctangents of Common Values-1
- Extra Example 2: Arcsin/Arccos/Arctan Values-2

19m 11s

- Intro0:00
- Pythagorean Identity0:17
- Pythagorean Triangle0:27
- Pythagorean Identity0:45
- Example 1: Use Pythagorean Theorem to Prove Pythagorean Identity1:14
- Example 2: Find Angle Given Cosine and Quadrant4:18
- Example 3: Verify Trigonometric Identity8:00
- Extra Example 1: Use Pythagorean Identity to Prove Pythagorean Theorem-1
- Extra Example 2: Find Angle Given Cosine and Quadrant-2

23m 16s

- Intro0:00
- Main Formulas0:19
- Companion to Pythagorean Identity0:27
- For Cotangents and Cosecants0:52
- How to Remember0:58
- Example 1: Prove the Identity1:40
- Example 2: Given Tan Find Sec3:42
- Example 3: Prove the Identity7:45
- Extra Example 1: Prove the Identity-1
- Extra Example 2: Given Sec Find Tan-2

52m 52s

- Intro0:00
- Addition and Subtraction Formulas0:09
- How to Remember0:48
- Cofunction Identities1:31
- How to Remember Graphically1:44
- Where to Use Cofunction Identities2:52
- Example 1: Derive the Formula for cos(A-B)3:08
- Example 2: Use Addition and Subtraction Formulas16:03
- Example 3: Use Addition and Subtraction Formulas to Prove Identity25:11
- Extra Example 1: Use cos(A-B) and Cofunction Identities-1
- Extra Example 2: Convert to Radians and use Formulas-2

29m 5s

- Intro0:00
- Main Formula0:07
- How to Remember from Addition Formula0:18
- Two Other Forms1:35
- Example 1: Find Sine and Cosine of Angle using Double Angle3:16
- Example 2: Prove Trigonometric Identity using Double Angle9:37
- Example 3: Use Addition and Subtraction Formulas12:38
- Extra Example 1: Find Sine and Cosine of Angle using Double Angle-1
- Extra Example 2: Prove Trigonometric Identity using Double Angle-2

43m 55s

- Intro0:00
- Main Formulas0:09
- Confusing Part0:34
- Example 1: Find Sine and Cosine of Angle using Half-Angle0:54
- Example 2: Prove Trigonometric Identity using Half-Angle11:51
- Example 3: Prove the Half-Angle Formula for Tangents18:39
- Extra Example 1: Find Sine and Cosine of Angle using Half-Angle-1
- Extra Example 2: Prove Trigonometric Identity using Half-Angle-2

25m 43s

- Intro0:00
- Master Formula for Right Angles0:11
- SOHCAHTOA0:15
- Only for Right Triangles1:26
- Example 1: Find All Angles in a Triangle2:19
- Example 2: Find Lengths of All Sides of Triangle7:39
- Example 3: Find All Angles in a Triangle11:00
- Extra Example 1: Find All Angles in a Triangle-1
- Extra Example 2: Find Lengths of All Sides of Triangle-2

56m 40s

- Intro0:00
- Law of Sines Formula0:18
- SOHCAHTOA0:27
- Any Triangle0:59
- Graphical Representation1:25
- Solving Triangle Completely2:37
- When to Use Law of Sines2:55
- ASA, SAA, SSA, AAA2:59
- SAS, SSS for Law of Cosines7:11
- Example 1: How Many Triangles Satisfy Conditions, Solve Completely8:44
- Example 2: How Many Triangles Satisfy Conditions, Solve Completely15:30
- Example 3: How Many Triangles Satisfy Conditions, Solve Completely28:32
- Extra Example 1: How Many Triangles Satisfy Conditions, Solve Completely-1
- Extra Example 2: How Many Triangles Satisfy Conditions, Solve Completely-2

49m 5s

- Intro0:00
- Law of Cosines Formula0:23
- Graphical Representation0:34
- Relates Sides to Angles1:00
- Any Triangle1:20
- Generalization of Pythagorean Theorem1:32
- When to Use Law of Cosines2:26
- SAS, SSS2:30
- Heron's Formula4:49
- Semiperimeter S5:11
- Example 1: How Many Triangles Satisfy Conditions, Solve Completely5:53
- Example 2: How Many Triangles Satisfy Conditions, Solve Completely15:19
- Example 3: Find Area of a Triangle Given All Side Lengths26:33
- Extra Example 1: How Many Triangles Satisfy Conditions, Solve Completely-1
- Extra Example 2: Length of Third Side and Area of Triangle-2

27m 37s

- Intro0:00
- Master Right Triangle Formula and Law of Cosines0:19
- SOHCAHTOA0:27
- Law of Cosines1:23
- Heron's Formula2:22
- Semiperimeter S2:37
- Example 1: Area of Triangle with Two Sides and One Angle3:12
- Example 2: Area of Triangle with Three Sides6:11
- Example 3: Area of Triangle with Three Sides, No Heron's Formula8:50
- Extra Example 1: Area of Triangle with Two Sides and One Angle-1
- Extra Example 2: Area of Triangle with Two Sides and One Angle-2

34m 25s

- Intro0:00
- Formulas to Remember0:11
- SOHCAHTOA0:15
- Law of Sines0:55
- Law of Cosines1:48
- Heron's Formula2:46
- Example 1: Telephone Pole Height4:01
- Example 2: Bridge Length7:48
- Example 3: Area of Triangular Field14:20
- Extra Example 1: Kite Height-1
- Extra Example 2: Roads to a Town-2

46m 42s

- Intro0:00
- Vector Formulas and Concepts0:12
- Vectors as Arrows0:28
- Magnitude0:38
- Direction0:50
- Drawing Vectors1:16
- Uses of Vectors: Velocity, Force1:37
- Vector Magnitude Formula3:15
- Vector Direction Formula3:28
- Vector Components6:27
- Example 1: Magnitude and Direction of Vector8:00
- Example 2: Force to a Box on a Ramp12:25
- Example 3: Plane with Wind18:30
- Extra Example 1: Components of a Vector-1
- Extra Example 2: Ship with a Current-2

1h 7m 35s

- Intro0:00
- Polar Coordinates vs Rectangular/Cartesian Coordinates0:12
- Rectangular Coordinates, Cartesian Coordinates0:23
- Polar Coordinates0:59
- Converting Between Polar and Rectangular Coordinates2:06
- R2:16
- Theta2:48
- Example 1: Convert Rectangular to Polar Coordinates6:53
- Example 2: Convert Polar to Rectangular Coordinates17:28
- Example 3: Graph the Polar Equation28:00
- Extra Example 1: Convert Polar to Rectangular Coordinates-1
- Extra Example 2: Graph the Polar Equation-2

35m 59s

- Intro0:00
- Main Definition0:07
- Number i0:23
- Complex Number Form0:33
- Powers of Imaginary Number i1:00
- Repeating Pattern1:43
- Operations on Complex Numbers3:30
- Adding and Subtracting Complex Numbers3:39
- Multiplying Complex Numbers4:39
- FOIL Method5:06
- Conjugation6:29
- Dividing Complex Numbers7:34
- Conjugate of Denominator7:45
- Example 1: Solve For Complex Number z11:02
- Example 2: Expand and Simplify15:34
- Example 3: Simplify the Powers of i17:50
- Extra Example 1: Simplify-1
- Extra Example 2: All Complex Numbers Satisfying Equation-2

40m 43s

- Intro0:00
- Polar Coordinates0:49
- Rectangular Form0:52
- Polar Form1:25
- R and Theta1:51
- Polar Form Conversion2:27
- R and Theta2:35
- Optimal Values4:05
- Euler's Formula4:25
- Multiplying Two Complex Numbers in Polar Form6:10
- Multiply r's Together and Add Exponents6:32
- Example 1: Convert Rectangular to Polar Form7:17
- Example 2: Convert Polar to Rectangular Form13:49
- Example 3: Multiply Two Complex Numbers17:28
- Extra Example 1: Convert Between Rectangular and Polar Forms-1
- Extra Example 2: Simplify Expression to Polar Form-2

57m 37s

- Intro0:00
- Introduction to DeMoivre's Theorem0:10
- n nth Roots3:06
- DeMoivre's Theorem: Finding nth Roots3:52
- Relation to Unit Circle6:29
- One nth Root for Each Value of k7:11
- Example 1: Convert to Polar Form and Use DeMoivre's Theorem8:24
- Example 2: Find Complex Eighth Roots15:27
- Example 3: Find Complex Roots27:49
- Extra Example 1: Convert to Polar Form and Use DeMoivre's Theorem-1
- Extra Example 2: Find Complex Fourth Roots-2

For more information, please see full course syllabus of Trigonometry

1 answer

Last reply by: Dr. William Murray

Fri Nov 3, 2017 6:37 PM

Post by Julius Francis on November 1, 2017

Do you recommend any Trigonometry Workbooks?

1 answer

Last reply by: Dr. William Murray

Tue Jul 18, 2017 8:32 AM

Post by Mohamed E Sowaileh on July 10, 2017

Hello Dr. William Murray,

I hope you are very well.

I am a student who is extremely weak in math. In order to be very strong in math, specially for engineering field, could you provide me with sequential order of mathematical topics and textbooks. With what should I begin so that I can master big topics like calculus, statistics, probability ... etc.

Your guidance is precious to me.

Thank you so much.

1 answer

Last reply by: Dr. William Murray

Sat Mar 18, 2017 11:56 PM

Post by Anna Kopituk on March 15, 2017

I am taking this as a home school student. How can I be tested on this material and come up with a grade?

1 answer

Last reply by: Dr. William Murray

Mon Jan 9, 2017 10:20 AM

Post by Anish Srinivasan on January 8, 2017

Is there a complement and supplement of a negative angle, and can there be a complement angle of an angle greater than 90 degrees? Also, can there be a supplement angle of an angle greater than 180 degrees?

3 answers

Last reply by: Dr. William Murray

Mon Nov 7, 2016 11:39 AM

Post by Firebird wang on November 2, 2016

Professor, I know that AP Statistics is not your subject, but I just wonder if you are able to watch the two videos which called Practice Test 2013 AP Statistics an Practice Test 2014 AP Statistics in the AP Statistics content? Both videos showing network error, I dont know why. I already tried in different computers already.

2 answers

Last reply by: Dr. William Murray

Mon Jun 13, 2016 8:53 PM

Post by Tiffany Warner on June 10, 2016

Hello Dr. Murray,

I see lots of comments regarding practice problems and such, but I see no link for them like I did with other lectures. Your examples in the video were definitely helpful for making the concepts sink in, but I do like to do practice problems every once in awhile to test myself and see if I really got it. Did they take them away?

Thank you!

1 answer

Last reply by: Dr. William Murray

Fri Oct 30, 2015 4:23 PM

Post by Alexander Roland on October 30, 2015

Hello Professor,

If you don't mind sharing, what type of technology is that you are using to deliver instructions?

Thanks for sharing

4 answers

Last reply by: Dr. William Murray

Wed Jun 17, 2015 10:18 AM

Post by Ashley Haden on April 29, 2015

It seems a bit confusing that radians are 2PIR. Why isn't there a symbol for a radian, or just for 2PI?

1 answer

Last reply by: Dr. William Murray

Mon Aug 4, 2014 7:31 PM

Post by Tehreem Lughmani on July 9, 2014

Example 3 how to know what is between 0-2pi? I'm not good with fractions :)

1 answer

Last reply by: Dr. William Murray

Mon Aug 4, 2014 7:13 PM

Post by Tehreem Lughmani on July 9, 2014

Example 3 - C. -586+360= -226+360= 134???

-226+360= 94

I get this answer every single time~ what's wrong here -.-|||

1 answer

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Wed Oct 9, 2013 5:48 PM

Post by Rakshit Joshi on October 7, 2013

How to download the notes??

1 answer

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Wed Oct 9, 2013 5:47 PM

Post by Rakshit Joshi on October 6, 2013

Sir you are AWESOME!!

1 answer

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Wed Aug 14, 2013 12:53 PM

Post by Reema Batra on August 1, 2013

I found errors in questions 6 and 7 in the practice problems...

1 answer

Last reply by: Dr. William Murray

Wed Aug 14, 2013 12:53 PM

Post by Reema Batra on August 1, 2013

For the sixth practice problem, I found an error:

Question - Determine which quadrant the following angle is in and find a coterminal angle between 0 and 360: 450.

My Answer: 90; y-axis.

The Answer Given: 115; Quadrant 2. This also had 450-360 is 115. This is incorrect, if I am not mistaken...

1 answer

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Fri Jul 5, 2013 9:57 AM

Post by mohammad sawari on July 5, 2013

what is call the half of the radius

1 answer

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Fri Jul 5, 2013 9:55 AM

Post by Norman Cervantes on July 1, 2013

second time going through this course. going straight to the examples, this course is very well taught. it really gives my brain a workout!

1 answer

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Mon Jun 10, 2013 7:34 PM

Post by Dr. Son's Statistics Class on June 10, 2013

You're a great professor!

1 answer

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Mon Jun 10, 2013 7:33 PM

Post by Jorge Sardinas on June 8, 2013

i am 9

3 answers

Last reply by: Dr. William Murray

Sat Jun 8, 2013 5:44 PM

Post by Manfred Berger on May 29, 2013

Could you elaborate a bit on what the motivation behind using signed angles is? Quite frankly I fail to see a functional difference between an angle -x and x+180 degrees.

3 answers

Last reply by: Dr. William Murray

Wed May 29, 2013 11:17 AM

Post by Manfred Berger on May 28, 2013

Your Rs look a lot like exponents. Is that a general notation or just your handwriting?

1 answer

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Thu Apr 25, 2013 3:04 PM

Post by Edmund Mercado on April 15, 2012

A very fine presentation.

1 answer

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Thu Apr 25, 2013 3:02 PM

Post by Levi Stafford on March 19, 2012

commenting on the text on the quick notes.

"2Ï€ parts, denoted 2Ï€ R." the pi's look like "n's" and it is confusing...I thought they were variables.

1 answer

Last reply by: Dr. William Murray

Thu Apr 25, 2013 2:59 PM

Post by kirill frusin on March 2, 2012

I 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 answer

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Thu Apr 25, 2013 2:53 PM

Post by Janet Wyatt on February 10, 2012

Is there practice worksheets I can print?

1 answer

Last reply by: Dr. William Murray

Thu Apr 25, 2013 2:50 PM

Post by Valtio Cooper on January 14, 2012

Great 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 answer

Last reply by: Dr. William Murray

Thu Apr 25, 2013 2:18 PM

Post by Kyle Spicer on December 6, 2011

where do you take the assessment test? I can't find it.

1 answer

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Thu Apr 25, 2013 2:16 PM

Post by Robert Reynolds on October 22, 2011

Thumbs 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 answer

Last reply by: Dr. William Murray

Thu Apr 25, 2013 2:15 PM

Post by David Burns on August 8, 2011

I wish this site had tests available, or at least links to them. Other than that I love it here.

1 answer

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Thu Apr 25, 2013 2:12 PM

Post by Sheila Greenfield on March 3, 2011

i get this and i'm a freshman in high school i really like this cant wait to learn more

1 answer

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Thu Apr 25, 2013 2:10 PM

Post by Erin Murphy on March 16, 2010

You are a fantastic prof. On to my next lesson!