Angles, Trigonometry - Educator.com

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	1 answer Last reply by: Dr. William Murray 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!
	1 answer Last reply by: Dr. William Murray 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 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 Last reply by: Dr. William Murray 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 2:18 PM Post by Kyle Spicer on December 6, 2011 where do you take the assessment test? I can't find it.
	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 2:53 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 2:59 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 Last reply by: Dr. William Murray Thu Apr 25, 2013 3:02 PM Post by Levi Stafford on March 19, 2012 commenting 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 answer Last reply by: Dr. William Murray Thu Apr 25, 2013 3:04 PM Post by Edmund Mercado on April 15, 2012 A 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 1^R 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 1^R 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:

Convert 27^° into radians.
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.

1000^°
− (19π /6)^R
-586^°
(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.

− (5π /4)^R
735^°
− (7π /3)^R
-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]

30^° and [(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

[(37π)/36]

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^°

126^°

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

108^°

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

[(19π)/45]

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^°

115^° is in Quadrant II

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

[(17π)/15] is in Quadrant III

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]

a. 90^° − 47^° = 43^°
b. [(π)/2] − [(π)/12] = [(5π)/12]

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]

a. 66^°
b. [(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

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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

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

Degrees

Radians

Converting Between Degrees and Radians

Formulas for Degrees and Radians

Coterminal, Complementary, Supplementary Angles

Example 1: Dividing a Circle

Example 2: Converting Between Degrees and Radians

Example 3: Quadrants and Coterminal Angles

Extra Example 1: Common Angle Conversions

Extra Example 2: Quadrants and Coterminal Angles

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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Angles

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Angles

A circle is divided into 12 equal angles. Calculate the measure of each angle in degrees and radians.

30^° and [(π)/6]

Convert 185^° into radians

[(37π)/36]

Convert [(7π)/10] into degrees

126^°

What is the degree measure of an arc whose measure is [(3π)/5] radians?

108^°

What is the radian measure of an arc whose measure is 76^°?

[(19π)/45]

Determine which quadrant the following angle is in and find a coterminal angle between 0^° and 360^° : 450^°

115^° is in Quadrant II

Determine which quadrant the following angle is in and find a coterminal angle between 0^° and 360^° : [( − 13π)/5]

[(17π)/15] is in Quadrant III

Find the complementary angle for each of the following angles:
a. 47^°
b. [(π)/12]

a. 90^° − 47^° = 43^°
b. [(π)/2] − [(π)/12] = [(5π)/12]

Find the supplementary angle for each of the following angles:
a. 114^°
b. [(4π)/5]

a. 66^°
b. [(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. [(16π)/11]; III
b. 263^°; III
c. [(25π)/13]; IV
d. 132^°; II

Answer

Angles

Mathematics: Trigonometry

Related Books

Related Books

Educator

the best free way to manage your classroom

Free Class Website!

Free Unlimited Hosting!

Free Educator.com Domain!

Angles

Share this knowledge with your friends!

Angles

A circle is divided into 12 equal angles. Calculate the measure of each angle in degrees and radians.

30° and [(π)/6]

Convert 185° into radians

[(37π)/36]

Convert [(7π)/10] into degrees

126°

What is the degree measure of an arc whose measure is [(3π)/5] radians?

108°

What is the radian measure of an arc whose measure is 76 °?

[(19π)/45]

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

115° is in Quadrant II

Determine which quadrant the following angle is in and find a coterminal angle between 0° and 360° : [( − 13π)/5]

[(17π)/15] is in Quadrant III

Find the complementary angle for each of the following angles: a. 47° b. [(π)/12]

a. 90° − 47° = 43° b. [(π)/2] − [(π)/12] = [(5π)/12]

Find the supplementary angle for each of the following angles: a. 114° b. [(4π)/5]

a. 66° b. [(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. [(16π)/11]; III b. 263°; III c. [(25π)/13]; IV d. 132°; II

Answer

Angles

Mathematics: Trigonometry

Related Books

Related Books

30^° and [(π)/6]

Convert 185^° into radians

126^°

108^°

What is the radian measure of an arc whose measure is 76^°?

Determine which quadrant the following angle is in and find a coterminal angle between 0^° and 360^° : 450^°

115^° is in Quadrant II

Determine which quadrant the following angle is in and find a coterminal angle between 0^° and 360^° : [( − 13π)/5]

Find the complementary angle for each of the following angles:
a. 47^°
b. [(π)/12]

a. 90^° − 47^° = 43^°
b. [(π)/2] − [(π)/12] = [(5π)/12]

Find the supplementary angle for each of the following angles:
a. 114^°
b. [(4π)/5]

a. 66^°
b. [(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. [(16π)/11]; III
b. 263^°; III
c. [(25π)/13]; IV
d. 132^°; II