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Professor Murray

Sine and Cosine Functions

Slide Duration:Table of Contents

I. Trigonometric Functions

Angles

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

Sine and Cosine Functions

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

Sine and Cosine Values of Special Angles

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

Modified Sine Waves: Asin(Bx+C)+D and Acos(Bx+C)+D

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

Tangent and Cotangent Functions

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

Secant and Cosecant Functions

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

Inverse Trigonometric Functions

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

Computations of Inverse Trigonometric Functions

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

II. Trigonometric Identities

Pythagorean Identity

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

Identity Tan(squared)x+1=Sec(squared)x

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

Addition and Subtraction Formulas

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

Double Angle Formulas

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

Half-Angle Formulas

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

III. Applications of Trigonometry

Trigonometry in Right Angles

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

Law of Sines

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

Law of Cosines

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

Finding the Area of a Triangle

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

Word Problems and Applications of Trigonometry

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

Vectors

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

IV. Complex Numbers and Polar Coordinates

Polar Coordinates

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

Complex Numbers

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

Polar Form of Complex Numbers

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

DeMoivre's Theorem

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

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For more information, please see full course syllabus of Trigonometry

For more information, please see full course syllabus of Trigonometry

Next Lecture

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1 answer

Last reply by: Dr. William Murray

Tue Jun 7, 2016 12:31 PM

Post by Bassam Razzaq on June 5, 2016

In example 2, how can the graph has value of x-coordinate more than 2pi. I thought 2pi was the last point as also seen in unit circle.

1 answer

Last reply by: Dr. William Murray

Mon Nov 2, 2015 2:18 PM

Post by Peter Ke on October 30, 2015

In example 4, I DON'T get how g(x) is odd. I thought it was even.

3 answers

Last reply by: Dr. William Murray

Wed Nov 11, 2015 9:23 AM

Post by Peter Ke on October 30, 2015

For example 3 shouldn't Sin = 4/5 be 3/5? Because 3 is the opposite and 4 is adjacent.

1 answer

Last reply by: Dr. William Murray

Thu Feb 19, 2015 3:36 PM

Post by patrick guerin on February 17, 2015

On practice question 2 it said the sin of theta was sqroot of 63 over 12 when i thought it was 9 over 12. Could you check it out please. Thanks.

1 answer

Last reply by: Dr. William Murray

Fri Dec 19, 2014 9:31 AM

Post by katrina williams on December 17, 2014

In the second to last practice problem what amount does n represent? I was able to draw the original graph but got lost by how far to move it over.

1 answer

Last reply by: Dr. William Murray

Mon Aug 4, 2014 7:48 PM

Post by patrick guerin on July 16, 2014

You placed a theta in the triangle that you created when you were defining sine cosine. Could you give me a small explanation on what theta is?

1 answer

Last reply by: Dr. William Murray

Tue Jun 17, 2014 12:08 PM

Post by Austin Cunningham on June 9, 2014

How come at around 8:00, he says that x^2 is the same thing as f(x)?

3 answers

Last reply by: Dr. William Murray

Thu Jun 5, 2014 11:43 AM

Post by Govind Balaji Srinivasa Raghavan on May 29, 2014

I dont know why. But all videos pause after sometime. Then it restarts again instead of continue. Also I cant skip to other part of video. Suppose tonight I watch half of video and go to sleep. Tomorrow morning, I have to watch from first, if i click on the play-head from where I should see, It automatically restarts.

1 answer

Last reply by: Dr. William Murray

Tue Mar 4, 2014 5:00 PM

Post by Damien O Byrne on February 28, 2014

Does sin cos and tan formulas only apply to right angled triangles

1 answer

Last reply by: Dr. William Murray

Wed Jan 22, 2014 3:04 PM

Post by Carroll Fields on January 16, 2014

In extra example I, at 3:30, why is the 'sine, cosine, and tangent of the right angle (alpha) , 1,0 , and undefined?And why for sine is it sin pi/2 ?

1 answer

Last reply by: Dr. William Murray

Mon Oct 21, 2013 7:15 PM

Post by yannick Haberkorn on October 12, 2013

i have to congratulate you Dr.william murray because i actually really feel i am in a learning environment and it feels great . Much thanks

1 answer

Last reply by: Dr. William Murray

Tue Apr 16, 2013 8:35 PM

Post by Dr. William Murray on January 27, 2013

Hi Emily,

Good question. As Jacob says in his post above, it's because we know that the graph of f(x-c) is like the graph of f(x), but shifted c units to the right. But you have to have the negative sign in there for this to work, so when we have f(x+(something)), we write it as f(x-(-something)). Then it's clear that the shift is (-something) units to the right, that is, (something) units to the left.

It's also worth reading Jacob's answer above -- same basic idea, but sometimes having a different person's phrasing helps.

Thanks for taking trigonometry!

Will Murray

1 answer

Last reply by: Dr. William Murray

Tue Apr 16, 2013 8:34 PM

Post by Emily Engle on January 27, 2013

At 28:10 Why do you change Sin (x+ Pi/2) to Sin (x-(- Pi/2)) ?

1 answer

Last reply by: Dr. William Murray

Fri Aug 31, 2012 5:45 PM

Post by Andraa Cram on June 25, 2012

@ 21:18, why, when going in the negative direction while graphing for sine (in red), does he draw the graph as (-Pi/2,-1) instead of (-Pi/2,1)? I'm very confused by this.

1 answer

Last reply by: Dr. William Murray

Sun Jan 27, 2013 4:18 PM

Post by Lourdes Johnson on June 3, 2012

Why does the lecture restart around a quarter in?

1 answer

Last reply by: Dr. William Murray

Sun Jan 27, 2013 4:16 PM

Post by Callistus Elue on May 23, 2012

the lecture reverts to the beginning almost as soon as it starts

1 answer

Last reply by: Dr. William Murray

Sun Jan 27, 2013 4:12 PM

Post by Jacob Burley on April 25, 2011

At 26:33 Professor Murray gave the algebraic equation for a graph that has a constant which was f(x-c). Our equation sin(x+pi/2) has a positive where the negative is in the original equation. In order to get the correct sign there we must change the + sign into two - signs because two negatives make a positive.

I know I'm not the greatest at explaining things but hopefully this helps a little bit.

1 answer

Last reply by: Dr. William Murray

Sun Jan 27, 2013 4:10 PM

Post by Shannon Bryington on February 28, 2011

At 27:40 on the video: Why was x + pi over 2 changed to x - neg pi over 2?

1 answer

Last reply by: Dr. William Murray

Sun Jan 27, 2013 3:59 PM

Post by Santhini Dheenathayalan on January 19, 2011

Great!