# Trigonometry Sine and Cosine Values of Special Angles

Section 1: Trigonometric Functions: Lecture 3 | 33:05 min

Special triangles are 45-45-90 and 30-60-90 and their side lengths have special proportions that we learn by heart. To find the sine and cosine values of special angles (angles that have 0°, 30°, 45°, 60°, 90°), we use the unit circle which has a radius of one so it’s easy to determine these values using trigonometry. We find the sine and the cosine of all the other angles using these ones from the first quadrant. The trick is to remember which ones are positive in which quadrant. The mnemonic ASTC (All Students Take Calculus) helps us remember that.

Professor Murray

Sine and Cosine Values of Special Angles

Slide Duration: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. Will Murray

Fri Jun 12, 2020 10:48 AM

Post by Jenny J. Hui on June 10 at 04:04:27 PM

This is really interesting and helpful. "All Students Take Calculus"!! YAY Thank you so much for making this course. I am not very good at Trigonometry and this helps a lot and I find myself getting interested in these topics. Thank you, sir.

1 answer

Last reply by: Dr. William Murray

Wed May 23, 2018 11:59 AM

Post by Harry Zhang on May 21, 2018

I love your lectures and find them really helpful, but when will we learn about other things, such as tangent, cotangent, secant, and cosecant? Sure, they sound like parts of a circle in a triangle, but I still really want to learn about those functions, so please tell me where I can find videos on those. Thank you very much.

1 answer

Last reply by: Dr. William Murray

Fri Jan 8, 2016 1:23 PM

Post by sania sarwar on January 7, 2016

Hi Sir

thanks for the lectures they are really helpful but can you please suggest me on which lecture to watch for this problem because I cant get my head around it.

If sin(theta)=0.3,cos(x)=0.7 and tan (alpha)=0.4 then find sin(3pi/2+theta)

1 answer

Last reply by: Dr. William Murray

Thu Feb 19, 2015 5:27 PM

Post by patrick guerin on February 19, 2015

Why is it that when you type in the cosine or sine of a number on a calculator, you get something like .0679 or something?

3 answers

Last reply by: Dr. William Murray

Mon Nov 17, 2014 8:12 PM

Post by Tami Cummins on September 20, 2013

I know this is so simple but I really do not understand how 5pi/3 equals pi + 2pi/3.

1 answer

Last reply by: Dr. William Murray

Fri Jun 21, 2013 6:22 PM

Post by A C on June 17, 2013

Please help: why, if I use a calculator to find the angle value of sin/cos/tan, does sin and tan give me the angle (neg. or pos. depending on the quadrant obviously) but cos gives me are larger angle (>90degrees), what I assume is the standard position angle?

1 answer

Last reply by: Dr. William Murray

Wed May 22, 2013 3:25 PM

Post by Monis Mirza on May 17, 2013

hi,

In the extra example I, how did you know that the special triangle is

45-45-90 triangle?

Thanks

1 answer

Last reply by: Dr. William Murray

Thu Nov 15, 2012 6:21 PM

Post by peter chrysanthopoulos on November 14, 2012

nevermind I got it

1 answer

Last reply by: Dr. William Murray

Thu Nov 15, 2012 6:20 PM

Post by peter chrysanthopoulos on November 14, 2012

how did you know that it was a 30/60/90 triangle in example 1?

1 answer

Last reply by: Dr. William Murray

Thu Apr 18, 2013 11:34 AM

Post by Dr. William Murray on October 17, 2012

Hi Chin,

It depends on which direction you're going. If it's radians to degrees, multiply by 180/pi. If it's degrees to radians, multiply by pi/180.

This makes sense if you follow the rules that you learn in physics and chemistry about units: 180 degrees = pi radians, so (180 degrees)/(pi radians) = 1. Then when you want to convert in either direction, you multiply by 1:

(3 pi/4 radians) x (180 degrees)/(pi radians) = 135 degrees.

90 degrees x (pi radians)/(180 degrees) = pi/2 radians.

Hope this helps. Thanks for studying trigonometry!

Will Murray

1 answer

Last reply by: Dr. William Murray

Thu Apr 18, 2013 11:35 AM

Post by chin chang on October 15, 2012

On the example problems you converted the degrees or radians into one or the other by multiplying pi/180 or 180/pi. How do you know what to multiply for each situation? Does it make sense? For instance on the second example problem, how did you to multiply 5pi/3 radians by 180/pi?

3 answers

Last reply by: Dr. William Murray

Fri Sep 28, 2012 4:54 PM

Post by Ivon Nieto Ivon Nieto on September 25, 2012

Is there an easier way to memorize the unit circle sine/cosine?