unit 12 trigonometry homework 4 answer key

• Tips for Success
• Information
• Google Classroom
• 8.1 Representing Data
• 8.2 Ratios and Rates
• 8.3 Pythagorean Relationship
• 8.4 Understanding Percent
• Unit 1 Project
• 8.5 Surface Area
• 8.6 Fraction Operations
• 8.8 Integers
• 8.9 Linear Relations
• 8.10 Solving Equations
• 8.11 Probability
• 10.6 Linear Relations and Functions
• 10.7 Linear Equations and Graphs
• 10.8 Systems of Equations
• 10.1 Measurement
• 10.2 Surface Area and Volume
• 10.3 Trigonometry
• 10.4 Exponents and Radicals
• 10.5 Polynomials
• Trigonometry
• 12.1 Functions and Transformations
• 12.2 Radical Functions
• 12.3 Polynomial Functions
• 12.4 Unit Circle Trigonometry
• 12.5 Graphs of Trigonometric Functions
• 12.6 Equations and Identities
• 12.7 Exponential Functions and Equations
• 12.8 Logarithmic Functions and Equations
• 12.9 Rational Functions and Equations
• 12.10 Function Operations
• 12.11 Permutations and Combinations
• Limits and Continuity
• Derivatives of Rational and Trigonometric Functions
• Derivatives of Exponential, Logarithmic and Inverse Functions
• Applications of Derivatives
• 12.5 Differentiable Equations
• 8. Velocity
• 9. Acceleration
• 12 Continental Drift
• 4.Atomic Theory
• 5.Classifying Compounds
• 6.Chemical Reactions
• 7.Radioactivity
• Planning Phase
• Game Interface

12.4.1a Angles and Radian Measure

• Ch4 Unit Outline
• p175 #1-6, 12-15, 17, 19, C1, C2, C3

Things you should be able to do after today:

• know the relationship between radians, the radius and the arc length in a circle
• convert degrees to radian measure
• convert radians to degrees
• know the special angles in degrees and radians 

12.4.1b Coterminal Angles

Angles in standard position use the positive side of the x-axis as the inital arm.  Counter-clockwise rotations from this initial arm produce positive angles, and clockwise rotations produce negative angles.  Sometimes, different angles can end up with the same terminal arm.  These are called coterminal angles.

Assignment:

• p175 #7-9, 11, 18-20 C5

Things you should know how to do after today:

• define an angle in standard position
• define a coterminal angle
• Find the reference angle for an angle in standard position
• Find the smallest positive coterminal angle for any angle in standard position

12.4.2a The Unit Circle

The foundation for a lot of trigonometry is the unit circle.

• p187 #1-4, 10, 11
• recall the relationship between radian measure with the number of radiuses around the circumference of a circle
• using special triangles, determine the coordinates for a point on the terminal arm when given the measure of the a standard position angle in radians

12.4.2b Unit Circle

• p187 #5, 6, 7, 9, 12, 13, 15, C1 C2 *17 *18
• Given the coordinates of a point on the unit circle, determine the measure of the standard position angle in radians

12.4.3a Trigonometric Ratios

• Example 2, DEF from the the notes will be marked next class
• p201 #1-9, 13, 14, 17, 18 C3
• sinθ, cosθ, tanθ, secθ, cscθ and cotθ

12.4.3b Finding the Arc Length

• Notes: 12.4.3b
• p201 #10, 11, 12, 15, 16, C1, C2 *20,22, 24

12.4.4 Solving Trigonometric Equations

• Notes: 4.4a
• Notes: 4.4b
• p211 #1, 3, 5, 6, 10, 11, 15, 16
• specific domains
• general solutions (solved over the reals)

12.4.5 Review

Warmup Answer Key:

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• Unit 1 - Algebraic Essentials Review
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Trigonometry (Algebra 2 Curriculum - Unit 12) | All Things Algebra®

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Due to the length of this Trigonometry Unit Bundle , it is divided into two parts with two unit tests. In addition to the unit tests, each part includes guided notes, homework assignments, quizzes, and study guides to cover the following topics:

Unit 12 Part I:

• Pythagorean Theorem

• Special Right Triangles

• Trigonometric Functions (sin, cos, tan, csc, sec, cot)

• Finding Side and Angle Measures

• Applications: Angle of Elevation and Depression

• Angles in Standard Position

• Converting between Degrees and Radians

• Coterminal and Reference Angles

• Trigonometric Functions in the Coordinate Plane

• The Unit Circle

• Law of Sines

• Law of Cosines

• Area of Triangles

• Applications of Law of Sines, Law of Cosines, and Area

Unit 12 Part II:

• Graphing Trigonometric Functions

• Trigonometric Identities

• Sum and Difference of Angle Identities

• Double-Angle and Half-Angle Identities

• Solving Trigonometric Equations

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Unit 4 – Solving Quadratics and Complex Numbers

Unit 5 – Polynomial Functions

Unit 6 – Radical Functions

Unit 7 – Exponential and Logarithmic Functions

Unit 8 – Rational Functions

Unit 9 – Conic Sections

Unit 10 – Sequences and Series

Unit 11 – Probability and Statistics

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3 π 2 3 π 2

−135 ° −135 °

7 π 10 7 π 10

α = 150° α = 150°

β = 60° β = 60°

7 π 6 7 π 6

215 π 18 = 37.525 units 215 π 18 = 37.525 units

− 3 π 2 − 3 π 2 rad/s

1655 kilometers per hour

7.2 Right Triangle Trigonometry

sin  t = 33 65 , cos  t = 56 65 , tan  t = 33 56 , sec  t = 65 56 , csc  t = 65 33 , cot  t = 56 33 sin  t = 33 65 , cos  t = 56 65 , tan  t = 33 56 , sec  t = 65 56 , csc  t = 65 33 , cot  t = 56 33

sin ( π 4 ) = 2 2 , cos ( π 4 ) = 2 2 , tan ( π 4 ) = 1 , sec ( π 4 ) = 2 , csc ( π 4 ) = 2 , cot ( π 4 ) = 1 sin ( π 4 ) = 2 2 , cos ( π 4 ) = 2 2 , tan ( π 4 ) = 1 , sec ( π 4 ) = 2 , csc ( π 4 ) = 2 , cot ( π 4 ) = 1

adjacent = 10 ; opposite = 10 3 ; adjacent = 10 ; opposite = 10 3 ; missing angle is π 6 π 6

About 52 ft

7.3 Unit Circle

cos ( t ) = − 2 2 , sin ( t ) = 2 2 cos ( t ) = − 2 2 , sin ( t ) = 2 2

cos ( π ) = − 1 , sin ( π ) = 0 cos ( π ) = − 1 , sin ( π ) = 0

sin ( t ) = − 7 25 sin ( t ) = − 7 25

approximately 0.866025403

• ⓐ cos ( 315° ) = 2 2 , sin ( 315° ) = – 2 2 cos ( 315° ) = 2 2 , sin ( 315° ) = – 2 2
• ⓑ cos ( − π 6 ) = 3 2 , sin ( − π 6 ) = − 1 2 cos ( − π 6 ) = 3 2 , sin ( − π 6 ) = − 1 2

( 1 2 , − 3 2 ) ( 1 2 , − 3 2 )

7.4 The Other Trigonometric Functions

sin t = − 2 2 cos t = 2 2 , tan t = − 1 , s e c t = 2 , csc t = − 2 , cot t = − 1 sin t = − 2 2 cos t = 2 2 , tan t = − 1 , s e c t = 2 , csc t = − 2 , cot t = − 1

sin π 3 = 3 2 , cos π 3 = 1 2 , tan π 3 = 3 , s e c π 3 = 2 , c s c π 3 = 2 3 3 , c o t π 3 = 3 3 sin π 3 = 3 2 , cos π 3 = 1 2 , tan π 3 = 3 , s e c π 3 = 2 , c s c π 3 = 2 3 3 , c o t π 3 = 3 3

sin ( − 7 π 4 ) = 2 2 , cos ( − 7 π 4 ) = 2 2 , tan ( − 7 π 4 ) = 1 , sec ( − 7 π 4 ) = 2 , csc ( − 7 π 4 ) = 2 , cot ( − 7 π 4 ) = 1 sin ( − 7 π 4 ) = 2 2 , cos ( − 7 π 4 ) = 2 2 , tan ( − 7 π 4 ) = 1 , sec ( − 7 π 4 ) = 2 , csc ( − 7 π 4 ) = 2 , cot ( − 7 π 4 ) = 1

sin t sin t

cos t = − 8 17 , sin t = 15 17 , tan t = − 15 8 csc t = 17 15 , cot t = − 8 15 cos t = − 8 17 , sin t = 15 17 , tan t = − 15 8 csc t = 17 15 , cot t = − 8 15

sin t = − 1 , cos t = 0 , tan t = Undefined sec t = Undefined, csc t = − 1 , cot t = 0 sin t = − 1 , cos t = 0 , tan t = Undefined sec t = Undefined, csc t = − 1 , cot t = 0

sec t = 2 , csc t = 2 , tan t = 1 , cot t = 1 sec t = 2 , csc t = 2 , tan t = 1 , cot t = 1

≈ − 2.414 ≈ − 2.414

7.1 Section Exercises

Whether the angle is positive or negative determines the direction. A positive angle is drawn in the counterclockwise direction, and a negative angle is drawn in the clockwise direction.

Linear speed is a measurement found by calculating distance of an arc compared to time. Angular speed is a measurement found by calculating the angle of an arc compared to time.

4 π 3 4 π 3

2 π 3 2 π 3

7 π 2 ≈ 11.00 in 2 7 π 2 ≈ 11.00 in 2

81 π 20 ≈ 12.72 cm 2 81 π 20 ≈ 12.72 cm 2

π 2 π 2 radians

−3 π −3 π radians

π π radians

5 π 6 5 π 6 radians

5.02 π 3 ≈ 5.26 5.02 π 3 ≈ 5.26 miles

25 π 9 ≈ 8.73 25 π 9 ≈ 8.73 centimeters

21 π 10 ≈ 6.60 21 π 10 ≈ 6.60 meters

104.7198 cm 2

0.7697 in 2

8 π 9 8 π 9

1320 1320 rad/min 210.085 210.085 RPM

7 7 in./s, 4.77 RPM , 28.65 28.65 deg/s

1 , 809 , 557.37 mm/min = 1 , 809 , 557.37 mm/min = 30.16 m/s 30.16 m/s

5.76 5.76 miles

794 miles per hour

2,234 miles per hour

11.5 inches

7.2 Section Exercises

The tangent of an angle is the ratio of the opposite side to the adjacent side.

For example, the sine of an angle is equal to the cosine of its complement; the cosine of an angle is equal to the sine of its complement.

b = 20 3 3 , c = 40 3 3 b = 20 3 3 , c = 40 3 3

a = 10,000 , c = 10,00.5 a = 10,000 , c = 10,00.5

b = 5 3 3 , c = 10 3 3 b = 5 3 3 , c = 10 3 3

5 29 29 5 29 29

5 41 41 5 41 41

c = 14 , b = 7 3 c = 14 , b = 7 3

a = 15 , b = 15 a = 15 , b = 15

b = 9.9970 , c = 12.2041 b = 9.9970 , c = 12.2041

a = 2.0838 , b = 11.8177 a = 2.0838 , b = 11.8177

a = 55.9808 , c = 57.9555 a = 55.9808 , c = 57.9555

a = 46.6790 , b = 17.9184 a = 46.6790 , b = 17.9184

a = 16.4662 , c = 16.8341 a = 16.4662 , c = 16.8341

498.3471 ft

22.6506 ft  

368.7633 ft

7.3 Section Exercises

The unit circle is a circle of radius 1 centered at the origin.

Coterminal angles are angles that share the same terminal side. A reference angle is the size of the smallest acute angle, t , t , formed by the terminal side of the angle t t and the horizontal axis.

The sine values are equal.

60° , 60° , Quadrant IV, sin ( 300° ) = − 3 2 sin ( 300° ) = − 3 2 , cos ( 300° ) = 1 2 cos ( 300° ) = 1 2

45° , 45° , Quadrant II, sin ( 135° ) = 2 2 sin ( 135° ) = 2 2 , cos ( 135° ) = − 2 2 cos ( 135° ) = − 2 2

60° , 60° , Quadrant II, sin ( 120° ) = 3 2 sin ( 120° ) = 3 2 , cos ( 120° ) = − 1 2 cos ( 120° ) = − 1 2

30° , 30° , Quadrant II, sin ( 150° ) = 1 2 sin ( 150° ) = 1 2 , cos ( 150° ) = − 3 2 cos ( 150° ) = − 3 2

π 6 , π 6 , Quadrant III, sin ( 7 π 6 ) = − 1 2 sin ( 7 π 6 ) = − 1 2 , cos ( 7 π 6 ) = − 3 2 cos ( 7 π 6 ) = − 3 2

π 4 , π 4 , Quadrant II, sin ( 3 π 4 ) = 2 2 sin ( 3 π 4 ) = 2 2 , cos ( 4 π 3 ) = − 2 2 cos ( 4 π 3 ) = − 2 2

π 3 , π 3 , Quadrant II, sin ( 2 π 3 ) = 3 2 sin ( 2 π 3 ) = 3 2 , cos ( 2 π 3 ) = − 1 2 cos ( 2 π 3 ) = − 1 2

π 4 , π 4 , Quadrant IV, sin ( 7 π 4 ) = − 2 2 , cos ( 7 π 4 ) = 2 2 sin ( 7 π 4 ) = − 2 2 , cos ( 7 π 4 ) = 2 2

− 15 4 − 15 4

( −10 , 10 3 ) ( −10 , 10 3 )

( –2.778 ,   15.757 ) ( –2.778 ,   15.757 )

[ –1 ,   1 ] [ –1 ,   1 ]

sin t = 1 2 , cos t = − 3 2 sin t = 1 2 , cos t = − 3 2

sin t = − 2 2 , cos t = − 2 2 sin t = − 2 2 , cos t = − 2 2

sin t = 3 2 , cos t = − 1 2 sin t = 3 2 , cos t = − 1 2

sin t = − 2 2 , cos t = 2 2 sin t = − 2 2 , cos t = 2 2

sin t = 0 , cos t = − 1 sin t = 0 , cos t = − 1

sin t = − 0.596 , cos t = 0.803 sin t = − 0.596 , cos t = 0.803

sin t = 1 2 , cos t = 3 2 sin t = 1 2 , cos t = 3 2

sin t = − 1 2 , cos t = 3 2 sin t = − 1 2 , cos t = 3 2

sin t = 0.761 , cos t = − 0.649 sin t = 0.761 , cos t = − 0.649

sin t = 1 , cos t = 0 sin t = 1 , cos t = 0

− 6 4 − 6 4

( 0 , –1 ) ( 0 , –1 )

37.5 seconds, 97.5 seconds, 157.5 seconds, 217.5 seconds, 277.5 seconds, 337.5 seconds

7.4 Section Exercises

Yes, when the reference angle is π 4 π 4 and the terminal side of the angle is in quadrants I and III. Thus, a x = π 4 , 5 π 4 , x = π 4 , 5 π 4 , the sine and cosine values are equal.

Substitute the sine of the angle in for y y in the Pythagorean Theorem x 2 + y 2 = 1. x 2 + y 2 = 1. Solve for x x and take the negative solution.

The outputs of tangent and cotangent will repeat every π π units.

2 3 3 2 3 3

− 2 3 3 − 2 3 3

− 3 3 − 3 3

sin t = − 2 2 3 sin t = − 2 2 3 , sec t = − 3 sec t = − 3 , csc t = − 3 2 4 csc t = − 3 2 4 , tan t = 2 2 tan t = 2 2 , cot t = 2 4 cot t = 2 4

sec t = 2 , sec t = 2 , csc t = 2 3 3 , csc t = 2 3 3 , tan t = 3 , tan t = 3 , cot t = 3 3 cot t = 3 3

− 2 2 − 2 2

sin t = 2 2 sin t = 2 2 , cos t = 2 2 cos t = 2 2 , tan t = 1 tan t = 1 , cot t = 1 cot t = 1 , sec t = 2 sec t = 2 , csc t = 2 csc t = 2

sin t = − 3 2 sin t = − 3 2 , cos t = − 1 2 cos t = − 1 2 tan t = 3 tan t = 3 , cot t = 3 3 cot t = 3 3 , sec t = − 2 sec t = − 2 , csc t = − 2 3 3 csc t = − 2 3 3

sin ( t ) ≈ 0.79 sin ( t ) ≈ 0.79

csc t ≈ 1.16 csc t ≈ 1.16

sin t cos t = tan t sin t cos t = tan t

13.77 hours, period: 1000 π 1000 π

3.46 inches

Review Exercises

− 7 π 6 − 7 π 6

10.385 meters

2 π 11 2 π 11

1036.73 miles per hour

a = 10 3 , c = 2 106 3 a = 10 3 , c = 2 106 3

a = 5 3 2 , b = 5 2 a = 5 3 2 , b = 5 2

369.2136 ft

all real numbers

cosine, secant

Practice Test

6.283 centimeters

3.351 feet per second, 2 π 75 2 π 75 radians per second

a = 9 2 , b = 9 3 2 a = 9 2 , b = 9 3 2

real numbers

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• Authors: Jay Abramson
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• Book title: Algebra and Trigonometry
• Publication date: Feb 13, 2015
• Location: Houston, Texas
• Book URL: https://openstax.org/books/algebra-and-trigonometry/pages/1-introduction-to-prerequisites
• Section URL: https://openstax.org/books/algebra-and-trigonometry/pages/chapter-7

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