Sine and Cosine Values of Special Angles, Trigonometry

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	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?
	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?
	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 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 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 Wed May 22, 2013 3:25 PM Post by Monis Mirza on May 17 at 01:14:01 PM hi, In the extra example I, how did you know that the special triangle is 45-45-90 triangle? Thanks

Sine and Cosine Values of Special Angles

Main definitions and formulas:

A 45-45-90 triangle has side lengths in proportion to 1-1-√ 2.
A 30-60-90 triangle has side lengths in proportion to 1-√ 3-2.
Degrees
Radians
Cosine
Sine

0
0
1
0

30
π
6

√ 3
2

1
2

45
π
4

√ 2
2

√ 2
2

60
π
3

1
2

√ 3
2

90
π
2

0
1
Use these values to find sines and cosines in other quadrants. The mnemonic ASTC (All Students Take Calculus) helps you remember which ones are positive in which quadrant. (All, Sine, Tangent, Cosine)

Example 1:

Convert 120^° to radians, identify its quadrant, and find its cosine and sine.

Example 2:

Convert (5π/3)^R to degrees, identify its quadrant, and find its cosine and sine.

Example 3:

Identify all angles between 0 and 2π whose sine is − [1/2], in both degrees and radians, and identify which quadrant each is in.

Example 4:

Convert 225^° to radians, identify its quadrant, and find its cosine and sine.

Example 5:

Identify all angles between 0 and 2π whose cosine is − (√3/2), in both degrees and radians, and identify which quadrant each is in.

Convert 315^° to radians, identify its quadrant, and find its cosine and sine.

First convert 315^° to radians by using the following formula: degree measure ×[(π)/(180^° )] = radian measure
315^° ×[(π)/(180^° )] = [(315^°π)/(180^° )] = [(7π)/4]
Locate [(7π)/4] on a unit circle (draw a picture). This will help you to determine the quadrant it is in and it will also help you to find the sine and cosine values.
Looking at our picture, we see that [(7π)/4] is in quadrant IV. This will help us to solve for the sine and cosine values as well.
Note that 315^° is 45^° from the x - axis so we can use the 45^° − 45^° − 90^° triangle to find the sine and cosine.
Recall that cosine is the x coordinate and sine is the y coordinate of [(7π)/4] on the unit circle
cos [(7π)/4] = [(√2 )/2] and sin [(7π)/4] = − [(√2 )/2]
Using the Mnemonic ASTC (All Students Take Calculus) helps to determine which values are positive in which quadrant
Since [(7π)/4] is in quadrant IV we know that cosine will be positive and sine will be negative

[(7π)/4] is in quadrant IV and cos [(7π)/4] = [(√2 )/2] and sin [(7π)/4] = − [(√2 )/2]

Convert [(4π)/3] to degrees, identify its quadrant, and find its cosine and sine.

First convert [(4π)/3] to degrees by using the following formula: radian measure ×[(180^°)/(π)] = degree measure
[(4π)/3] ×[(180^°)/(π)] = 4 ×60^° = 240^°
Locate 240^° on a unit circle (draw a picture). This will help you to determine the quadrant it is in and it will also help you to find the sine and cosine values.
Looking at our picture, we see that 240^° is in quadrant III. This will help us to solve for the sine and cosine values as well.
Note that [(4π)/3] is [(π)/3] past the x - axis so we can use the 30^° − 60^° − 90^° triangle to find the sine and cosine.
Recall that cosine is the x coordinate and sine is the y coordinate of 240^° on the unit circle
cos 240^° = − [1/2] and sin 240^° = − [(√3 )/2]
Using the Mnemonic ASTC (All Students Take Calculus) helps to determine which values are positive in which quadrant
Since 240^° is in quadrant III we know that tangent will be positive so sine and cosine will be negative

240^° is in quadrant III and cos 240^° = − [1/2] and sin 240^° = − [(√3 )/2]

Convert 210^° to radians, identify its quadrant, and find its cosine and sine.

First convert 210^° to radians by using the following formula: degree measure ×[(π)/(180^°)] = radian measure
210^° ×[(π)/(180^°)] = [(210^°π)/(180^°)] = [(7π)/6]
Locate [(7π)/6] on a unit circle (draw a picture). This will help you to determine the quadrant it is in and it will also help you to find the sine and cosine values.
Looking at our picture, we see that [(7π)/6] is in quadrant III. This will help us to solve for the sine and cosine values as well.
Note that 210^° is 30^° past the x - axis so we can use the 30^° − 60^° − 90^° triangle to find the sine and cosine.
Recall that cosine is the x coordinate and sine is the y coordinate of [(7π)/6] on the unit circle
cos [(7π)/6] = − [(√3 )/2] and sin [(7π)/6] = − [1/2]
Using the Mnemonic ASTC (All Students Take Calculus) helps to determine which values are positive in which quadrant
Since [(7π)/6] is in quadrant III we know that tangent will be positive so cosine and sine will be negative

[(7π)/6] is in quadrant III and cos [(7π)/6] = − [(√3 )/2] and sin [(7π)/6] = − [1/2]

Convert [(3π)/4] to degrees, identify its quadrant, and find its cosine and sine.

First convert [(3π)/4] to degrees by using the following formula: radian measure ×[(180^°)/(π)] = degree measure
[(3π)/4] ×[(180^°)/(π)] = 3 ×45^° = 135^°
Locate 135^° on a unit circle (draw a picture). This will help you to determine the quadrant it is in and it will also help you to find the sine and cosine values.
Looking at our picture, we see that 135^° is in quadrant II. This will help us to solve for the sine and cosine values as well.
Note that [(3π)/4] is [(π)/4] from the x - axis so we can use the 45^° − 45^° − 90° triangle to find the sine and cosine.
Recall that cosine is the x coordinate and sine is the y coordinate of 135^° on the unit circle
cos 135^° = − [(√2 )/2] and sin 135^° = [(√2 )/2]
Using the Mnemonic ASTC (All Students Take Calculus) helps to determine which values are positive in which quadrant
Since 135^° is in quadrant II we know that sine will be positive and cosine will be negative

135^° is in quadrant II and cos 135^° = − [(√2 )/2] and sin 135^° = [(√2 )/2]

Convert 150^° to radians, identify its quadrant, and find its cosine and sine.

First convert 150^° to radians by using the following formula: degree measure ×[(π)/(180^°)] = radian measure
150^° ×[(π)/(180^°)] = [(150^°π)/(180^°)] = [(5π)/6]
Locate [(5π)/6] on a unit circle (draw a picture). This will help you to determine the quadrant it is in and it will also help you to find the sine and cosine values.
Looking at our picture, we see that [(5π)/6] is in quadrant II. This will help us to solve for the sine and cosine values as well.
Note that 150^° is 30^° from the x - axis so we can use the 30^° - 60^° - 90^° triangle to find the sine and cosine.
Recall that cosine is the x coordinate and sine is the y coordinate of [(5π)/6] on the unit circle
cos [(5π)/6] = − [(√3 )/2] and sin [(5π)/6] = [1/2]
Using the Mnemonic ASTC (All Students Take Calculus) helps to determine which values are positive in which quadrant
Since [(5π)/6] is in quadrant II we know that sine will be positive and cosine will be negative

[(5π)/6] is in quadrant II and cos [(5π)/6] = − [(√3 )/2] and sin [(5π)/6] = [1/2]

Evaluate sin([(3π)/4])

Locate [(3π)/4] on a unit circle (draw a picture). This will help you to determine the quadrant it is in and it will also help you to find the sine value.
Looking at our picture, we see that [(3π)/4] is in quadrant II. This will help us to solve for the sine value.
Note that [(3π)/4] is [(π)/4] from the x - axis so we can use the 45^° − 45^° − 90^° triangle to find the sine value.
Recall that sine is the y coordinate of [(3π)/4] on the unit circle
sin [(3π)/4] = [(√2 )/2]
Using the Mnemonic ASTC (All Students Take Calculus) helps to determine which values are positive in which quadrant
Since [(3π)/4] is in quadrant II we know that sine will be positive.

sin ([(3π)/4]) = [(√2 )/2]

Evaluate cos 330^°

Locate 330^° on a unit circle (draw a picture). This will help you to determine the quadrant it is in and it will also help you to find the cosine value.
Looking at our picture, we see that 330^° is in quadrant IV. This will help us to solve for the cosine value.
Note that 330^° is 30^° from the x - axis so we can use the 30^° - 60^° - 90^° triangle to find the cosine value.
Recall that cosine is the x coordinate of 330^° on the unit circle
cos 330^° = [(√3 )/2]
Using the Mnemonic ASTC (All Students Take Calculus) helps to determine which values are positive in which quadrant
Since 330^° is in quadrant IV we know that cosine will be positive.

cos 330^° = [(√3 )/2]

Evaluate sin([(5π)/4])

Locate [(5π)/4] on a unit circle (draw a picture). This will help you to determine the quadrant it is in and it will also help you to find the sine value.
Looking at our picture, we see that [(5π)/4] is in quadrant III. This will help us to solve for the sine value.
Note that [(5π)/4] is [(π)/4] past the x - axis so we can use the 45^° − 45^° − 90^° triangle to find the sine value.
Recall that sine is the y coordinate of [(5π)/4] on the unit circle
sin [(5π)/4] = − [(√2 )/2]
Using the Mnemonic ASTC (All Students Take Calculus) helps to determine which values are positive in which quadrant
Since [(5π)/4] is in quadrant III we know that sine will be negative.

sin ([(5π)/4]) = − [(√2 )/2]

Identify all angles between 0 and 2π whose cosine is [1/2] in both degrees and radians. Identify which quadrant each is in.

Recall that cosine is the x coordinate.
Draw a unit circle and locate positive [1/2] along the x - axis. Draw a vertical line through that point. This will help you to identify the two points on the circle.
Connect the points on the circle to the origin. Notice that a 30^° - 60^° - 90^° triangle was formed.
The first angle is 60^° past 0^° ⇒ 0^° + 60^° = 60^°
The second angle is 30^° past of 270^° ⇒ 270^° + 30^° = 300^°

Angle 1 Angle 2

Degree 60^° 300^°

Radian [(π)/3] [(5π)/3]

Quadrant I IV

Identify all angles between 0 and 2π whose sine is − [(√3 )/2] in both degrees and radians. Identify which quadrant each is in.

Recall that sine is the y coordinate.
Draw a unit circle and locate − [(√3 )/2] along the y - axis. Draw a horizontal line through that point. This will help you to identify the two points on the circle.
Connect the points on the circle to the origin. Notice that a 30^° - 60^° - 90^° triangle was formed.
The first angle is 60^° past 180^° ⇒ 180^° + 60^° = 240^°
The second angle is 60^° short of 360^° ⇒ 360^° − 60^° = 300^°

Angle 1 Angle 2

Degree 240^° 300^°

Radian [(4π)/3] [(5π)/3]

Quadrant I IV

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*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.

Answer

Sine and Cosine Values of Special 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.

Intro

45-45-90 Triangle and 30-60-90 Triangle

Mnemonic: All Students Take Calculus (ASTC)

Example 1: Convert, Quadrant, Sine/Cosine

Example 2: Convert, Quadrant, Sine/Cosine

Example 3: All Angles and Quadrants

Extra Example 1: Convert, Quadrant, Sine/Cosine

Extra Example 2: All Angles and Quadrants

I. Trigonometric Functions
	Angles	39:05
	Sine and Cosine Functions	43:16
	Sine and Cosine Values of Special Angles	33:05
	Modified Sine Waves: Asin(Bx+C)+D and Acos(Bx+C)+D	52:03
	Tangent and Cotangent Functions	36:04
	Secant and Cosecant Functions	27:18
	Inverse Trigonometric Functions	32:58
	Computations of Inverse Trigonometric Functions	31:08
II. Trigonometric Identities
	Pythagorean Identity	19:11
	Identity Tan(squared)x+1=Sec(squared)x	23:16
	Addition and Subtraction Formulas	52:52
	Double Angle Formulas	29:05
	Half-Angle Formulas	43:55
III. Applications of Trigonometry
	Trigonometry in Right Angles	25:43
	Law of Sines	56:40
	Law of Cosines	49:05
	Finding the Area of a Triangle	27:37
	Word Problems and Applications of Trigonometry	34:25
	Vectors	46:42
IV. Complex Numbers and Polar Coordinates
	Polar Coordinates	67:35
	Complex Numbers	35:59
	Polar Form of Complex Numbers	40:43
	DeMoivre's Theorem	57:37

Mathematics: Trigonometry

I. Trigonometric Functions
	Angles			39:05
		Intro		0:00
		Degrees		0:22
			Circle is 360 Degrees	0:48
			Splitting a Circle	1:13
		Radians		2:08
			Circle is 2 Pi Radians	2:31
			One Radian	2:52
			Half-Circle and Right Angle	4:00
		Converting Between Degrees and Radians		6:24
			Formulas for Degrees and Radians	6:52
		Coterminal, Complementary, Supplementary Angles		7:23
			Coterminal Angles	7:30
			Complementary Angles	9:40
			Supplementary Angles	10:08
		Example 1: Dividing a Circle		10:38
		Example 2: Converting Between Degrees and Radians		11:56
		Example 3: Quadrants and Coterminal Angles		14:18
		Extra Example 1: Common Angle Conversions		8:02
		Extra Example 2: Quadrants and Coterminal Angles		7:14
	Sine and Cosine Functions			43:16
		Intro		0:00
		Sine and Cosine		0:15
			Unit Circle	0:22
			Coordinates on Unit Circle	1:03
			Right Triangles	1:52
			Adjacent, Opposite, Hypotenuse	2:25
			Master Right Triangle Formula: SOHCAHTOA	2:48
		Odd Functions, Even Functions		4:40
			Example: Odd Function	4:56
			Example: Even Function	7:30
		Example 1: Sine and Cosine		10:27
		Example 2: Graphing Sine and Cosine Functions		14:39
		Example 3: Right Triangle		21:40
		Example 4: Odd, Even, or Neither		26:01
		Extra Example 1: Right Triangle		4:05
		Extra Example 2: Graphing Sine and Cosine Functions		5:23
	Sine and Cosine Values of Special Angles			33:05
		Intro		0:00
		45-45-90 Triangle and 30-60-90 Triangle		0:08
			45-45-90 Triangle	0:21
			30-60-90 Triangle	2:06
		Mnemonic: All Students Take Calculus (ASTC)		5:21
			Using the Unit Circle	5:59
			New Angles	6:21
			Other Quadrants	9:43
			Mnemonic: All Students Take Calculus	10:13
		Example 1: Convert, Quadrant, Sine/Cosine		13:11
		Example 2: Convert, Quadrant, Sine/Cosine		16:48
		Example 3: All Angles and Quadrants		20:21
		Extra Example 1: Convert, Quadrant, Sine/Cosine		4:15
		Extra Example 2: All Angles and Quadrants		4:03
	Modified Sine Waves: Asin(Bx+C)+D and Acos(Bx+C)+D			52:03
		Intro		0:00
		Amplitude and Period of a Sine Wave		0:38
			Sine Wave Graph	0:58
			Amplitude: Distance from Middle to Peak	1:18
			Peak: Distance from Peak to Peak	2:41
		Phase Shift and Vertical Shift		4:13
			Phase Shift: Distance Shifted Horizontally	4:16
			Vertical Shift: Distance Shifted Vertically	6:48
		Example 1: Amplitude/Period/Phase and Vertical Shift		8:04
		Example 2: Amplitude/Period/Phase and Vertical Shift		17:39
		Example 3: Find Sine Wave Given Attributes		25:23
		Extra Example 1: Amplitude/Period/Phase and Vertical Shift		7:27
		Extra Example 2: Find Cosine Wave Given Attributes		10:27
	Tangent and Cotangent Functions			36:04
		Intro		0:00
		Tangent and Cotangent Definitions		0:21
			Tangent Definition	0:25
			Cotangent Definition	0:47
		Master Formula: SOHCAHTOA		1:01
			Mnemonic	1:16
		Tangent and Cotangent Values		2:29
			Remember Common Values of Sine and Cosine	2:46
			90 Degrees Undefined	4:36
		Slope and Mnemonic: ASTC		5:47
			Uses of Tangent	5:54
			Example: Tangent of Angle is Slope	6:09
			Sign of Tangent in Quadrants	7:49
		Example 1: Graph Tangent and Cotangent Functions		10:42
		Example 2: Tangent and Cotangent of Angles		16:09
		Example 3: Odd, Even, or Neither		18:56
		Extra Example 1: Tangent and Cotangent of Angles		2:27
		Extra Example 2: Tangent and Cotangent of Angles		5:02
	Secant and Cosecant Functions			27:18
		Intro		0:00
		Secant and Cosecant Definitions		0:17
			Secant Definition	0:18
			Cosecant Definition	0:33
		Example 1: Graph Secant Function		0:48
		Example 2: Values of Secant and Cosecant		6:49
		Example 3: Odd, Even, or Neither		12:49
		Extra Example 1: Graph of Cosecant Function		4:58
		Extra Example 2: Values of Secant and Cosecant		5:19
	Inverse Trigonometric Functions			32:58
		Intro		0:00
		Arcsine Function		0:24
			Restrictions between -1 and 1	0:43
			Arcsine Notation	1:26
		Arccosine Function		3:07
			Restrictions between -1 and 1	3:36
			Cosine Notation	3:53
		Arctangent Function		4:30
			Between -Pi/2 and Pi/2	4:44
			Tangent Notation	5:02
		Example 1: Domain/Range/Graph of Arcsine		5:45
		Example 2: Arcsin/Arccos/Arctan Values		10:46
		Example 3: Domain/Range/Graph of Arctangent		17:14
		Extra Example 1: Domain/Range/Graph of Arccosine		4:30
		Extra Example 2: Arcsin/Arccos/Arctan Values		5:40
	Computations of Inverse Trigonometric Functions			31:08
		Intro		0:00
		Inverse Trigonometric Function Domains and Ranges		0:31
			Arcsine	0:41
			Arccosine	1:14
			Arctangent	1:41
		Example 1: Arcsines of Common Values		2:44
		Example 2: Odd, Even, or Neither		5:57
		Example 3: Arccosines of Common Values		12:24
		Extra Example 1: Arctangents of Common Values		5:50
		Extra Example 2: Arcsin/Arccos/Arctan Values		8:51
II. Trigonometric Identities
	Pythagorean Identity			19:11
		Intro		0:00
		Pythagorean Identity		0:17
			Pythagorean Triangle	0:27
			Pythagorean Identity	0:45
		Example 1: Use Pythagorean Theorem to Prove Pythagorean Identity		1:14
		Example 2: Find Angle Given Cosine and Quadrant		4:18
		Example 3: Verify Trigonometric Identity		8:00
		Extra Example 1: Use Pythagorean Identity to Prove Pythagorean Theorem		3:32
		Extra Example 2: Find Angle Given Cosine and Quadrant		3:55
	Identity Tan(squared)x+1=Sec(squared)x			23:16
		Intro		0:00
		Main Formulas		0:19
			Companion to Pythagorean Identity	0:27
			For Cotangents and Cosecants	0:52
			How to Remember	0:58
		Example 1: Prove the Identity		1:40
		Example 2: Given Tan Find Sec		3:42
		Example 3: Prove the Identity		7:45
		Extra Example 1: Prove the Identity		2:22
		Extra Example 2: Given Sec Find Tan		4:34
	Addition and Subtraction Formulas			52:52
		Intro		0:00
		Addition and Subtraction Formulas		0:09
			How to Remember	0:48
		Cofunction Identities		1:31
			How to Remember Graphically	1:44
			Where to Use Cofunction Identities	2:52
		Example 1: Derive the Formula for cos(A-B)		3:08
		Example 2: Use Addition and Subtraction Formulas		16:03
		Example 3: Use Addition and Subtraction Formulas to Prove Identity		25:11
		Extra Example 1: Use cos(A-B) and Cofunction Identities		7:54
		Extra Example 2: Convert to Radians and use Formulas		11:32
	Double Angle Formulas			29:05
		Intro		0:00
		Main Formula		0:07
			How to Remember from Addition Formula	0:18
			Two Other Forms	1:35
		Example 1: Find Sine and Cosine of Angle using Double Angle		3:16
		Example 2: Prove Trigonometric Identity using Double Angle		9:37
		Example 3: Use Addition and Subtraction Formulas		12:38
		Extra Example 1: Find Sine and Cosine of Angle using Double Angle		6:10
		Extra Example 2: Prove Trigonometric Identity using Double Angle		3:18
	Half-Angle Formulas			43:55
		Intro		0:00
		Main Formulas		0:09
			Confusing Part	0:34
		Example 1: Find Sine and Cosine of Angle using Half-Angle		0:54
		Example 2: Prove Trigonometric Identity using Half-Angle		11:51
		Example 3: Prove the Half-Angle Formula for Tangents		18:39
		Extra Example 1: Find Sine and Cosine of Angle using Half-Angle		7:16
		Extra Example 2: Prove Trigonometric Identity using Half-Angle		3:34
III. Applications of Trigonometry
	Trigonometry in Right Angles			25:43
		Intro		0:00
		Master Formula for Right Angles		0:11
			SOHCAHTOA	0:15
			Only for Right Triangles	1:26
		Example 1: Find All Angles in a Triangle		2:19
		Example 2: Find Lengths of All Sides of Triangle		7:39
		Example 3: Find All Angles in a Triangle		11:00
		Extra Example 1: Find All Angles in a Triangle		5:10
		Extra Example 2: Find Lengths of All Sides of Triangle		4:18
	Law of Sines			56:40
		Intro		0:00
		Law of Sines Formula		0:18
			SOHCAHTOA	0:27
			Any Triangle	0:59
			Graphical Representation	1:25
			Solving Triangle Completely	2:37
		When to Use Law of Sines		2:55
			ASA, SAA, SSA, AAA	2:59
			SAS, SSS for Law of Cosines	7:11
		Example 1: How Many Triangles Satisfy Conditions, Solve Completely		8:44
		Example 2: How Many Triangles Satisfy Conditions, Solve Completely		15:30
		Example 3: How Many Triangles Satisfy Conditions, Solve Completely		28:32
		Extra Example 1: How Many Triangles Satisfy Conditions, Solve Completely		8:01
		Extra Example 2: How Many Triangles Satisfy Conditions, Solve Completely		15:11
	Law of Cosines			49:05
		Intro		0:00
		Law of Cosines Formula		0:23
			Graphical Representation	0:34
			Relates Sides to Angles	1:00
			Any Triangle	1:20
			Generalization of Pythagorean Theorem	1:32
		When to Use Law of Cosines		2:26
			SAS, SSS	2:30
		Heron's Formula		4:49
			Semiperimeter S	5:11
		Example 1: How Many Triangles Satisfy Conditions, Solve Completely		5:53
		Example 2: How Many Triangles Satisfy Conditions, Solve Completely		15:19
		Example 3: Find Area of a Triangle Given All Side Lengths		26:33
		Extra Example 1: How Many Triangles Satisfy Conditions, Solve Completely		11:05
		Extra Example 2: Length of Third Side and Area of Triangle		9:17
	Finding the Area of a Triangle			27:37
		Intro		0:00
		Master Right Triangle Formula and Law of Cosines		0:19
			SOHCAHTOA	0:27
			Law of Cosines	1:23
		Heron's Formula		2:22
			Semiperimeter S	2:37
		Example 1: Area of Triangle with Two Sides and One Angle		3:12
		Example 2: Area of Triangle with Three Sides		6:11
		Example 3: Area of Triangle with Three Sides, No Heron's Formula		8:50
		Extra Example 1: Area of Triangle with Two Sides and One Angle		2:54
		Extra Example 2: Area of Triangle with Two Sides and One Angle		6:48
	Word Problems and Applications of Trigonometry			34:25
		Intro		0:00
		Formulas to Remember		0:11
			SOHCAHTOA	0:15
			Law of Sines	0:55
			Law of Cosines	1:48
			Heron's Formula	2:46
		Example 1: Telephone Pole Height		4:01
		Example 2: Bridge Length		7:48
		Example 3: Area of Triangular Field		14:20
		Extra Example 1: Kite Height		4:36
		Extra Example 2: Roads to a Town		10:34
	Vectors			46:42
		Intro		0:00
		Vector Formulas and Concepts		0:12
			Vectors as Arrows	0:28
			Magnitude	0:38
			Direction	0:50
			Drawing Vectors	1:16
			Uses of Vectors: Velocity, Force	1:37
			Vector Magnitude Formula	3:15
			Vector Direction Formula	3:28
			Vector Components	6:27
		Example 1: Magnitude and Direction of Vector		8:00
		Example 2: Force to a Box on a Ramp		12:25
		Example 3: Plane with Wind		18:30
		Extra Example 1: Components of a Vector		2:54
		Extra Example 2: Ship with a Current		13:13
IV. Complex Numbers and Polar Coordinates
	Polar Coordinates			67:35
		Intro		0:00
		Polar Coordinates vs Rectangular/Cartesian Coordinates		0:12
			Rectangular Coordinates, Cartesian Coordinates	0:23
			Polar Coordinates	0:59
		Converting Between Polar and Rectangular Coordinates		2:06
			R	2:16
			Theta	2:48
		Example 1: Convert Rectangular to Polar Coordinates		6:53
		Example 2: Convert Polar to Rectangular Coordinates		17:28
		Example 3: Graph the Polar Equation		28:00
		Extra Example 1: Convert Polar to Rectangular Coordinates		10:01
		Extra Example 2: Graph the Polar Equation		10:53
	Complex Numbers			35:59
		Intro		0:00
		Main Definition		0:07
			Number i	0:23
			Complex Number Form	0:33
		Powers of Imaginary Number i		1:00
			Repeating Pattern	1:43
		Operations on Complex Numbers		3:30
			Adding and Subtracting Complex Numbers	3:39
			Multiplying Complex Numbers	4:39
			FOIL Method	5:06
			Conjugation	6:29
		Dividing Complex Numbers		7:34
			Conjugate of Denominator	7:45
		Example 1: Solve For Complex Number z		11:02
		Example 2: Expand and Simplify		15:34
		Example 3: Simplify the Powers of i		17:50
		Extra Example 1: Simplify		4:37
		Extra Example 2: All Complex Numbers Satisfying Equation		10:00
	Polar Form of Complex Numbers			40:43
		Intro		0:00
		Polar Coordinates		0:49
			Rectangular Form	0:52
			Polar Form	1:25
			R and Theta	1:51
		Polar Form Conversion		2:27
			R and Theta	2:35
			Optimal Values	4:05
			Euler's Formula	4:25
		Multiplying Two Complex Numbers in Polar Form		6:10
			Multiply r's Together and Add Exponents	6:32
		Example 1: Convert Rectangular to Polar Form		7:17
		Example 2: Convert Polar to Rectangular Form		13:49
		Example 3: Multiply Two Complex Numbers		17:28
		Extra Example 1: Convert Between Rectangular and Polar Forms		6:48
		Extra Example 2: Simplify Expression to Polar Form		7:48
	DeMoivre's Theorem			57:37
		Intro		0:00
		Introduction to DeMoivre's Theorem		0:10
			n nth Roots	3:06
		DeMoivre's Theorem: Finding nth Roots		3:52
			Relation to Unit Circle	6:29
			One nth Root for Each Value of k	7:11
		Example 1: Convert to Polar Form and Use DeMoivre's Theorem		8:24
		Example 2: Find Complex Eighth Roots		15:27
		Example 3: Find Complex Roots		27:49
		Extra Example 1: Convert to Polar Form and Use DeMoivre's Theorem		7:41
		Extra Example 2: Find Complex Fourth Roots		14:36

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Sine and Cosine Values of Special Angles