unit 12 trigonometry homework 8 graphing trigonometric functions

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Unit 11: Trigonometry

About this unit.

Let's extend trigonometric ratios sine, cosine, and tangent into functions that are defined for all real numbers. You might be surprised at how we can use the behavior of those functions to model real-world situations involving carnival rides and planetary distances.

Unit circle introduction

• Unit circle (Opens a modal)
• The trig functions & right triangle trig ratios (Opens a modal)
• Trig unit circle review (Opens a modal)
• Unit circle Get 3 of 4 questions to level up!
• Intro to radians (Opens a modal)
• Radians & degrees (Opens a modal)
• Degrees to radians (Opens a modal)
• Radians to degrees (Opens a modal)
• Radian angles & quadrants (Opens a modal)
• Radians & degrees Get 3 of 4 questions to level up!
• Unit circle (with radians) Get 3 of 4 questions to level up!

The Pythagorean identity

• Proof of the Pythagorean trig identity (Opens a modal)
• Using the Pythagorean trig identity (Opens a modal)
• Pythagorean identity review (Opens a modal)
• Use the Pythagorean identity Get 3 of 4 questions to level up!

Trigonometric values of special angles

• Trig values of π/4 (Opens a modal)
• Trig values of special angles Get 3 of 4 questions to level up!

Graphs of sin(x), cos(x), and tan(x)

• Graph of y=sin(x) (Opens a modal)
• Intersection points of y=sin(x) and y=cos(x) (Opens a modal)
• Graph of y=tan(x) (Opens a modal)

Amplitude, midline and period

• Features of sinusoidal functions (Opens a modal)
• Midline, amplitude, and period review (Opens a modal)
• Midline of sinusoidal functions from graph Get 3 of 4 questions to level up!
• Amplitude of sinusoidal functions from graph Get 3 of 4 questions to level up!
• Period of sinusoidal functions from graph Get 3 of 4 questions to level up!

Transforming sinusoidal graphs

• Amplitude & period of sinusoidal functions from equation (Opens a modal)
• Transforming sinusoidal graphs: vertical stretch & horizontal reflection (Opens a modal)
• Transforming sinusoidal graphs: vertical & horizontal stretches (Opens a modal)
• Amplitude of sinusoidal functions from equation Get 3 of 4 questions to level up!
• Midline of sinusoidal functions from equation Get 3 of 4 questions to level up!
• Period of sinusoidal functions from equation Get 3 of 4 questions to level up!

Graphing sinusoidal functions

• Example: Graphing y=3⋅sin(½⋅x)-2 (Opens a modal)
• Example: Graphing y=-cos(π⋅x)+1.5 (Opens a modal)
• Sinusoidal function from graph (Opens a modal)
• Graph sinusoidal functions Get 3 of 4 questions to level up!
• Construct sinusoidal functions Get 3 of 4 questions to level up!
• Graph sinusoidal functions: phase shift Get 3 of 4 questions to level up!

Sinusoidal models

• Interpreting trigonometric graphs in context (Opens a modal)
• Trig word problem: modeling daily temperature (Opens a modal)
• Trig word problem: modeling annual temperature (Opens a modal)
• Trig word problem: length of day (phase shift) (Opens a modal)
• Trigonometry: FAQ (Opens a modal)
• Interpreting trigonometric graphs in context Get 3 of 4 questions to level up!
• Modeling with sinusoidal functions Get 3 of 4 questions to level up!
• Modeling with sinusoidal functions: phase shift Get 3 of 4 questions to level up!

Trigonometry (Algebra 2 Curriculum - Unit 12) | All Things Algebra®

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Description

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

ADDITIONAL COMPONENTS INCLUDED:

(1) Links to Instructional Videos: Links to videos of each lesson in the unit are included. Videos were created by fellow teachers for their students using the guided notes and shared in March 2020 when schools closed with no notice.  Please watch through first before sharing with your students. Many teachers still use these in emergency substitute situations. (2) Editable Assessments: Editable versions of each quiz and the unit test are included. PowerPoint is required to edit these files. Individual problems can be changed to create multiple versions of the assessment. The layout of the assessment itself is not editable. If your Equation Editor is incompatible with mine (I use MathType), simply delete my equation and insert your own.

(3) Google Slides Version of the PDF: The second page of the Video links document contains a link to a Google Slides version of the PDF. Each page is set to the background in Google Slides. There are no text boxes;  this is the PDF in Google Slides.  I am unable to do text boxes at this time but hope this saves you a step if you wish to use it in Slides instead! 

This resource is included in the following bundle(s):

Algebra 2 Curriculum

More Algebra 2 Units:

Unit 1 – Equations and Inequalities

Unit 2 – Linear Functions and Systems

Unit 3 – Parent Functions and Transformations

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

LICENSING TERMS: This purchase includes a license for one teacher only for personal use in their classroom. Licenses are non-transferable , meaning they can not be passed from one teacher to another. No part of this resource is to be shared with colleagues or used by an entire grade level, school, or district without purchasing the proper number of licenses. If you are a coach, principal, or district interested in transferable licenses to accommodate yearly staff changes, please contact me for a quote at [email protected].

COPYRIGHT TERMS: This resource may not be uploaded to the internet in any form, including classroom/personal websites or network drives, unless the site is password protected and can only be accessed by students.

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All Things Algebra®

Algebra 2 Unit 12: Trigonometry

This unit includes 96 pages of guided notes, homework assignments, three quizzes, two study guides, and two unit tests that cover the topics listed in the description below.

• Description
• Additional Information
• What Educators Are Saying

This unit contains the following topics:

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

Note:  Due to the length of this unit, it is divided into two parts with two unit tests.

This unit does not contain activities.

This is the guided notes, homework assignments, quizzes, study guide, and unit test only.  For suggested activities to go with this unit, check out the ATA Activity Alignment Guides .

This resource is included in the following bundle(s):

Algebra 2 Curriculum

License Terms:

This purchase includes a single non-transferable license, meaning it is for one teacher only for personal use in their classroom and can not be passed from one teacher to another.  No part of this resource is to be shared with colleagues or used by an entire grade level, school, or district without purchasing the proper number of licenses.  A t ransferable license is not available for this resource.

Copyright Terms:

No part of this resource may be uploaded to the internet in any form, including classroom/personal websites or network drives, unless the site is password protected and can only be accessed by students.

What standards is this curriculum aligned to?

What format are the files in, will i have access to materials if they are updated, are answer keys included, are videos included.

All things algebra is the best math creator in my opinion, I will continuously use and purchase the products! Students find them easy to use, engaging, and fun! I love the opportunities for differentiation as well.

I'm teaching a Pre-Calculus class but with students whose skill level is slightly below average (and most of whom won't be on a Calculus track for senior year/college), and using this Alg. II unit for the Trigonometry lessons has been really helpful. It breaks it down for them and covers all of the basics/necessities without getting too "into the weeds" on the parts that I knew they would struggle more with.

This is a great bridge to my textbook that is a little high for the average student. My lower level classes, this is the perfect fit in not set up and work. The topic examples are on target for the standards as well.

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11.3: Trigonometric Functions

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• Page ID 122920

• Katherine Yoshiwara
• Los Angeles Pierce College

4.1 Angles and Rotation

Homework 4.1.

a \(216^{\circ}\)

b \(108^{\circ}\)

c \(480^{\circ}\)

d \(960^{\circ}\)

a \(\dfrac{1}{8}\)

b \(\dfrac{5}{6}\)

c \(\dfrac{3}{2}\)

d \(\dfrac{7}{6}\)

a \(\dfrac{2}{3}\)

b \(\dfrac{5}{3}\)

7. \(60^{\circ}\)

9. \(60^{\circ}\)

11. \(14^{\circ}\)

13. \(400^{\circ} \text{ and } -320^{\circ}\) (Answers vary.)

15. \(575^{\circ} \text{ and } -145^{\circ}\) (Answers vary.)

17. \(665^{\circ} \text{ and } -55^{\circ}\) (Answers vary.)

19. \(295^{\circ}\)

21. \(70^{\circ}\)

23. \(315^{\circ}\)

25. \(80^{\circ}\)

27. \(36^{\circ}\)

29. \(63^{\circ}\)

31. \(165^{\circ}, 95^{\circ}, 345^{\circ}\)

33. \(140^{\circ}, 220^{\circ}, 320^{\circ}\)

35. \(112^{\circ}, 248^{\circ}, 292^{\circ}\)

37. −0.9205

39. −0.7193

43. −0.7193

a \(120^{\circ}\)

b \(135^{\circ}\)

c \(150^{\circ}\)

d \(210^{\circ}\)

e \(225^{\circ}\)

f \(240^{\circ}\)

g \(300^{\circ}\)

h \(315^{\circ}\)

i \(330^{\circ}\)

\begin{aligned} \mathrm{b} \sin 120^{\circ} & =\dfrac{\sqrt{3}}{2}, \cos 120^{\circ}=\dfrac{-1}{2}, \tan 120^{\circ}=-\sqrt{3}, \\ \sin 240^{\circ} & =\dfrac{-\sqrt{3}}{2}, \cos 240^{\circ}=\dfrac{-1}{2}, \tan 240^{\circ}=\sqrt{3}, \\ \sin 300^{\circ} & =\dfrac{-\sqrt{3}}{2}, \cos 300^{\circ}=\dfrac{1}{2}, \tan 300^{\circ}=-\sqrt{3} \end{aligned}

\begin{aligned} \text { b } \sin 135^{\circ} & =\dfrac{1}{\sqrt{2}}, \quad \cos 135^{\circ}=\dfrac{-1}{\sqrt{2}}, \quad \tan 135^{\circ}=-1, \\ \sin 225^{\circ} & =\dfrac{-1}{\sqrt{2}}, \quad \cos 225^{\circ}=\dfrac{-1}{\sqrt{2}}, \quad \tan 225^{\circ}=1, \\ \sin 315^{\circ} & =\dfrac{-1}{\sqrt{2}}, \quad \cos 315^{\circ}=\dfrac{1}{\sqrt{2}}, \quad \tan 315^{\circ}=-1 \end{aligned}

a III and IV

b II and III

c I and III

a \(0^{\circ}\) and \(180^{\circ}\)

b \(90^{\circ}\) and \(270^{\circ}\)

55. \(105^{\circ}\)

57. \(264^{\circ}\)

59. \(313^{\circ}\)

61. \(83^{\circ}, 263^{\circ}\)

63. \(23^{\circ}, 337^{\circ}\)

65. \(265^{\circ}, 275^{\circ}\)

67. \(156^{\circ}, 204^{\circ}\)

69. \(246^{\circ}, 294^{\circ}\)

71. \(149^{\circ}, 329^{\circ}\)

73. \((-2\sqrt{2}, 2\sqrt{2})\)

75. \((\dfrac{3}{2}, \dfrac{3\sqrt{3}}{2})\)

77. \((\dfrac{-\sqrt{3}}{2}, \dfrac{-1}{2})\)

a (−0.9, −0.3)

b (−0.940, −0.342)

c (−1.9, −0.7)

a (−0.9, 0.3)

b (−0.940, 0.342)

c (−1.9, 0.7)

83. Sides of similar triangles are proportional.

4.2 Graphs of Trigonometric Functions

Homework 4.2.

1. \((4 \sqrt{2},-4 \sqrt{2})\)

3. \((-10,-10 \sqrt{3})\)

5. \(\left(\dfrac{-15 \sqrt{3}}{2}, \dfrac{15}{2}\right)\)

7. \((-1.25,-5.87)\)

9. \((5.70,-11.86)\)

11. \((9.46,-3.26)\)

b 15.3 mi east, 21 mi north

b 91.9 km west, 77.1 km south

b 30.9 km west, 8.3 km north

a \(\left(-225^{\circ}, \dfrac{1}{\sqrt{2}}\right)\)

b \(\left(-135^{\circ}, \dfrac{-1}{\sqrt{2}}\right)\)

c \(\left(-90^{\circ},-1\right)\)

d \(\left(45^{\circ}, \dfrac{1}{\sqrt{2}}\right)\)

e \(\left(180^{\circ}, 0\right)\)

f \(\left(315^{\circ}, \dfrac{-1}{\sqrt{2}}\right)\)

a \(\left(-240^{\circ}, \dfrac{-1}{2}\right)\)

b \(\left(-210^{\circ}, \dfrac{-\sqrt{3}}{2}\right)\)

c \(\left(-60^{\circ}, \dfrac{-1}{2}\right)\)

d \(\left(30^{\circ}, \dfrac{\sqrt{3}}{2}\right)\)

e \(\left(120^{\circ}, \dfrac{-1}{2}\right)\)

f \(\left(270^{\circ}, 0\right)\)

33. \(\dfrac{7}{2}\)

35. \(-2\sqrt{2} - 1\)

39. \(\dfrac{21}{2}\)

a \(36.9^{\circ}, 143.1^{\circ}\)

a \(72.5^{\circ}, 287.5^{\circ}\)

49. \(36.9^{\circ}, 143.1^{\circ}\)

51. \(72.5^{\circ}, 287.5^{\circ}\)

53. \(191.5^{\circ}, 348.5^{\circ}\)

55. \(154.2^{\circ}, 205.8^{\circ}\)

b \(\tan \theta\) approaches \(\infty\)

d \(\tan \theta\) approaches \(-\infty\)

e The calculator gives an error message because \(\tan 90^{\circ}\) is undefined.

61. \(51.34^{\circ}\)

63. \(159.44^{\circ}\)

65. \(y+5=\left(\tan 28^{\circ}\right)(x-3)\) or \(y+5=0.532(x-3)\)

67. \(y-12=\left(\tan 112^{\circ}\right)(x+8)\) or \(y-12=-2.475(x+8)\)

a The slope increases toward \(\infty\).

b The slope decreases toward \(-\infty\).

4.3 Periodic Functions

Homework 4.3.

d The graph from \(t=24\) to \(t=48\) will be exactly the same shape as the graph from \(t=0\) to \(t=24\). \(f(t+24) = f(t)\) says that the ant's \(y\)-coordinate 24 seconds after a time \(t\) is the same as its \(y\)-coordinate at time \(t\).

a He will be back in the same position.

b \(f(d+40) = f(d)\)

c The graph for \(0 \leq d \leq 40\) will be exactly the same shape as the graph for \(40 \leq d \leq 80\).

d Every 40 unit wide piece of the graph will be identical to the previous 40 units.

7. \(y = 6 \sin \theta\)

9. \(y = \cos \theta - 5\)

11. \(y = \sin (4\theta)\)

19. amp = 4, period = \(360^{\circ}\), midline: \(y = 3\)

21. amp = 5, period = \(180^{\circ}\), midline: \(y = 0\)

23. amp = 3, period = \(120^{\circ}\), midline: \(y = -4\)

a amp = 1, period = \(90^{\circ}\), midline: \(y = 0\)

b \(y = \sin 4 \theta\)

a amp = 1, period = \(360^{\circ}\), midline: \(y = 3\)

b \(y = 3 + \cos \theta\)

a amp = 4, period = \(360^{\circ}\), midline: \(y = -2\)

b \(y = -2 + 4\sin \theta\)

a amp = 2, period = \(120^{\circ}\), midline: \(y = 2\)

b \(y = 2 + 2\cos 3 \theta\)

33. \(y = 2 + 5 \cos \theta\)

35. \(y = -4 \sin \theta\)

37. \(y = -4 + 6\sin 3 \theta\) (Answers vary)

39. \(y = 3 + 2 \cos \theta\) (Answers vary)

41. \(y = 12 \cos 2 \theta\)

43. \(A\left(0^{\circ},-3\right), B\left(135^{\circ}, \dfrac{3}{\sqrt{2}}\right), C\left(300^{\circ}, \dfrac{-3}{2}\right)\)

45. \(P\left(112.5^{\circ}, 1\right), Q\left(180^{\circ}, 0\right), R\left(337.5^{\circ},-1\right)\)

47. \(X\left(45^{\circ},-3+\dfrac{1}{\sqrt{2}}\right), Y\left(90^{\circ},-3\right) Z\left(300^{\circ},-2\right)\)

49. not periodic

51. Periodic with period 4

b 10 minutes

b period 1 sec, midline \(y = 12\), amp 10 inches

b period 1 year, midline \(y = 3500\), amp 2500

b period 1 year, midline \(y = 51\), amp 21

a Emotional high: Oct 5 and Nov 3, low: Oct 19; Physical high: Sep 30 and Oct 23, low: Oct 12 and Nov 4; Intellectual high: Oct 10, low: Oct 26

b Emotional: 28 days, physical: 23 days, intellectual: 32 days

c 5152 days

a periodic, period 8

b 4, midline: \(y = 3\)

c \(k = 8\)

d \(a = 3, b = 7\)

a systolic 120 mm Hg, diastolic 80 mm Hg, pulse pressure 40 mm Hg.

b \(99 \dfrac{1}{3}\)

c 72 beats per minute

a 69 hours.

b 2.2 to 3.5

c The larger dip corresponds to when the brighter star is eclipsed, the smaller dip corresponds to when the dimmer star is eclipsed.

4.4 Chapter 4 Summary and Review

Chapter 4 review problems.

1. \(12^{\circ}\)

a \(150^{\circ}, -210^{\circ}\)

b \(240^{\circ},-120^{\circ}\)

c \(160^{\circ},-560^{\circ}\)

d \(20^{\circ},-340^{\circ}\)

a \(I, 60^{\circ} ; 120^{\circ}, 240^{\circ}, 300^{\circ}\)

b \(I V, 25^{\circ} ; 155^{\circ}, 205^{\circ}, 335^{\circ}\)

c \(I I, 80^{\circ} ; 80^{\circ}, 260^{\circ}, 280^{\circ}\)

d \(I I I, 70^{\circ} ; 70^{\circ}, 110^{\circ}, 290^{\circ}\)

9. \(210^{\circ}, 330^{\circ}\)

11. \(120^{\circ}, 240^{\circ}\)

13. \(45^{\circ}, 225^{\circ}\)

15. \(23^{\circ}, 337^{\circ}\)

17. \(72^{\circ}, 252^{\circ}\)

19. \(163^{\circ}, 277^{\circ}\)

21. \(221.81^{\circ}, 318.19^{\circ}\)

23. \(123.69^{\circ}, 303.69^{\circ}\)

25. \(128.68^{\circ}, 231.32^{\circ}\)

27. (−9.74, −2.25)

29. (−0.28, 8.00)

31. (2.84, 0.98)

33. south: 1.74 mi, west: 9.85 mi

35. \(y=4+7 \sin (180 \theta)\)

37. \(y=17+7 \sin \theta\)

39. \(\dfrac{\sqrt{3}}{2}\)

43. \(y=1.5 \cos \left(\dfrac{\theta}{3}\right), M\left(-90^{\circ}, \dfrac{3 \sqrt{3}}{4}\right), N\left(180^{\circ}, \dfrac{3}{4}\right)\)

45. \(y=3+3 \sin 2 \theta, A\left(-45^{\circ}, 6\right), B\left(120^{\circ}, 3-\dfrac{3 \sqrt{3}}{2}\right)\)

b amp: 2, period: \(360^{\circ}\), midline: \(y = 4\)

b amp: 3.5, period: \(180^{\circ}\), midline: \(y = 1.5\)

55. \(30^{\circ}\)

57. \(92.05^{\circ}\)

59. \(y = x + 2\)

61. \(y = - \sqrt{3}x + 3 \sqrt{3} - 4\)

The \(\theta\)-intercepts of \(\cos \theta\) occur at the vertical asymptotes of \(\tan \theta\).

Chapter 8: More Functions and Identities

Exercises: 8.3 The Reciprocal Functions Exercises

Practice each skill in the Homework Problems listed.

• Evaluate the reciprocal trig functions for angles in degrees or radians #1–20
• Find values or expressions for the six trig ratios #21–28
• Evaluate the reciprocal trig functions in applications #29–32
• Given one trig ratio, find the others #33–46, 71–80
• Evaluate expressions exactly #47–52
• Graph the secant, cosecant, and cotangent functions #53–58
• Identify graphs of the reciprocal trig functions #59–64
• Solve equations in secant, cosecant, and cotangent #65–70
• Use identities to simplify or evaluate expressions #81–94

Suggested Homework

Exercises for 8.3 The Reciprocal Functions

Exercise group.

For Problems 1–8, evaluate. Round answers to 3 decimal places.

[latex]\csc 27°[/latex]

[latex]\sec 8°[/latex]

[latex]\cot 65°[/latex]

[latex]\csc 11°[/latex]

[latex]\sec 1.4[/latex]

[latex]\cot 4.3[/latex]

[latex]\csc \dfrac{5\pi}{16}[/latex]

[latex]\sec \dfrac{7\pi}{20}[/latex]

For Problems 9–16, evaluate. Give exact values.

[latex]\csc 30°[/latex]

[latex]\sec 0°[/latex]

[latex]\cot 45[/latex]

[latex]\csc 60°[/latex]

[latex]\sec 150°[/latex]

[latex]\cot 120°[/latex]

[latex]\csc 135°[/latex]

[latex]\sec 270°[/latex]

For Problems 17–18, complete the tables with exact values.

Evaluate. Round answers to three decimal places.

• [latex]\displaystyle \cos 0.2[/latex]
• [latex]\displaystyle (\cos 0.2)^{-1}[/latex]
• [latex]\displaystyle \cos^{-1}0.2[/latex]
• [latex]\displaystyle \dfrac{1}{\cos 0.2}[/latex]
• [latex]\displaystyle \cos\dfrac{1}{0.2}[/latex]
• [latex]\displaystyle \sec 0.2[/latex]
• [latex]\displaystyle \tan 3.2[/latex]
• [latex]\displaystyle \tan^{-1} 3.2[/latex]
• [latex]\displaystyle \cot 3.2[/latex]
• [latex]\displaystyle \dfrac{1}{\tan 3.2}[/latex]
• [latex]\displaystyle \tan \dfrac{1}{3.2}[/latex]
• [latex]\displaystyle (\tan 3.2)^{-1}[/latex]

For Problems 21–28, find exact values for the six trigonometric ratios of the angle [latex]\theta{.}[/latex]

• Write an expression for the distance, [latex]d{,}[/latex] that sunlight travels through a layer of atmosphere of thickness [latex]h{.}[/latex]
• Find the distance (to the nearest mile) that sunlight travels through a 100-mile layer of atmosphere when the sun is [latex]40°[/latex] above the horizon.
• Use the figure to write an expression for the radius, [latex]r{,}[/latex] of a curve whose degree of curvature is [latex]\theta{.}[/latex] (Hint: The bisector of the angle [latex]\theta[/latex] is perpendicular to the chord.)
• Find the radius of a curve whose degree of curvature is [latex]43°{.}[/latex]

When a plane is tilted by an angle [latex]\theta[/latex] from the horizontal, the time required for a ball starting from rest to roll a horizontal distance of [latex]l[/latex] feet on the plane is

[latex]t=\sqrt{\dfrac{l}{8}\csc(2\theta)}~~ {seconds}[/latex]

• How long, to the nearest 0.01 second, will it take the ball to roll 2 feet horizontally on a plane tilted by [latex]12°{?}[/latex]
• Solve the formula for [latex]l[/latex] in terms of [latex]t[/latex] and [latex]\theta{.}[/latex]

After a heavy rainfall, the depth, [latex]D{,}[/latex] of the runoff flow at a distance [latex]x[/latex] feet from the watershed down a slope at angle [latex]\alpha[/latex] is given by

[latex]D=(kx)^{0.6}(\cot \alpha)^{0.3}~~{inches}[/latex]

where [latex]k[/latex] is a constant determined by the surface roughness and the intensity of the runoff.

• How deep, to the nearest 0.01 inch, is the runoff 100 feet down a slope of [latex]10°[/latex] if [latex]k=0.0006{?}[/latex]
• Solve the formula for [latex]x[/latex] in terms of [latex]D[/latex] and [latex]\alpha{.}[/latex]

For Problems 33–38, write algebraic expressions for the six trigonometric ratios of the angle [latex]\theta{.}[/latex]

The diagram shows a unit circle. Find six line segments whose lengths are, respectively, [latex]\sin t,~ \cos t,~ \tan t,~ \sec t,~ \csc t,[/latex] and [latex]\cot t{.}[/latex]

Use the figure in Problem 39 to find each area in terms of the angle [latex]t{.}[/latex]

• [latex]\displaystyle \triangle OAC[/latex]
• [latex]\displaystyle \triangle OBD[/latex]
• sector [latex]OAC[/latex]
• [latex]\displaystyle \triangle OFE[/latex]

For Problems 41–46, sketch the reference angle and find exact values for all six trigonometric functions of the angle.

[latex]\sec \theta = 2,~~\theta[/latex] in Quadrant IV

[latex]\csc \phi = 4,~~\phi[/latex] in Quadrant II

[latex]\csc \alpha = 3,~~\alpha[/latex] in Quadrant I

[latex]\sec \beta = 4,~~\beta[/latex] in Quadrant IV

[latex]\cot \gamma = \dfrac{1}{4},~~\gamma[/latex] in Quadrant III

[latex]\tan \theta = 6,~~\theta[/latex] in Quadrant I

For Problems 47–52, evaluate.

[latex]4 \cot \dfrac{\pi}{3} + 2\sec \dfrac{\pi}{4}[/latex]

[latex]\dfrac{1}{2} \csc \dfrac{\pi}{6} - \dfrac{1}{4}\cot \dfrac{\pi}{6}[/latex]

[latex]\dfrac{1}{2} \csc \dfrac{5\pi}{3}\cot \dfrac{3\pi}{4}[/latex]

[latex]6 \cot \dfrac{7\pi}{6} \sec \dfrac{5\pi}{4}[/latex]

[latex]\left(\csc \dfrac{2\pi}{3} - \sec \dfrac{3\pi}{4}\right)^2[/latex]

[latex]\sec^2 \dfrac{5\pi}{6} \csc^2 \dfrac{4\pi}{3}[/latex]

Complete the table and sketch a graph of [latex]y=\sec x{.}[/latex]

Complete the table and sketch a graph of [latex]y=\csc x{.}[/latex]

Complete the table and sketch a graph of [latex]y=\cot x{.}[/latex]

For Problems 59-64,

• Graph each function in the ZTrig window and write a simpler expression for the function.
• Show algebraically that your new expression is equivalent to the original one.

[latex]y=\dfrac{\csc x}{\cot x}[/latex]

[latex]y=\dfrac{\sec x}{\tan x}[/latex]

[latex]y=\dfrac{\sec x \cot x}{\csc x}[/latex]

[latex]y=\dfrac{\csc x \tan x}{\sec x}[/latex]

[latex]y=\tan x \csc x[/latex]

[latex]y=\sin x \sec x[/latex]

For Problems 65–70, find all solutions between [latex]0[/latex] and [latex]2\pi{.}[/latex]

[latex]3\csc \theta + 2=8[/latex]

[latex]-2\sec \theta + 7=3[/latex]

[latex]\sqrt{2}\sec \theta =-2[/latex]

[latex]8+\csc \theta =6[/latex]

[latex]2\cot \theta = -\sqrt{12}[/latex]

[latex]\sqrt{3}\cot \theta =1[/latex]

For Problems 71–76, use identities to find exact values or to write algebraic expressions.

If [latex]\tan \alpha = -2[/latex] and [latex]\dfrac{\pi}{2} \lt \alpha \lt \pi{,}[/latex] find [latex]\cos \alpha{.}[/latex]

If [latex]\cot \beta = \dfrac{5}{4}[/latex] and [latex]\pi \lt \beta \lt \dfrac{3\pi}{2}{,}[/latex] find [latex]\sin \beta{.}[/latex]

If [latex]\sec x = \dfrac{a}{2}[/latex] and [latex]0 \lt \alpha \lt \dfrac{\pi}{2}{,}[/latex] find [latex]\tan x{.}[/latex]

If [latex]\csc y = \dfrac{1}{b}[/latex] and [latex]\dfrac{\pi}{2} \lt y \lt \pi{,}[/latex] find [latex]\cot y{.}[/latex]

If [latex]\csc \phi = w[/latex] and [latex]\dfrac{3\pi}{2} \lt \alpha \lt 2\pi{,}[/latex] find [latex]\cos \phi{.}[/latex]

If [latex]\sec \theta = \dfrac{3}{z}[/latex] and [latex]\pi \lt \alpha \lt \dfrac{\pi}{2}{,}[/latex] find [latex]\sin \theta{.}[/latex]

For Problems 77–80, find exact values for [latex]\sec s,~ \csc s,[/latex] and [latex]\cot s{.}[/latex]

For Problems 81–88, write the expression in terms of sine and cosine, and simplify.

[latex]\sec \theta \tan \theta[/latex]

[latex]\csc \phi \cot \phi[/latex]

[latex]\dfrac{\csc t}{cot t}[/latex]

[latex]\dfrac{\tan v}{\sec v}[/latex]

[latex]\sec \beta - \tan \beta[/latex]

[latex]\cot \alpha + \csc \alpha[/latex]

[latex]\sin x \tan x - \sec x[/latex]

[latex]\csc y - \cos y \cot y[/latex]

Prove the Pythagorean identity [latex]1 + \tan^2 \theta = \sec^2 \theta{.}[/latex] (Hint: Start with the identity [latex]\cos^2 \theta + \sin^2 \theta = 1[/latex] and divide both sides of the equation by [latex]\cos^2 \theta{.}[/latex])

Prove the Pythagorean identity [latex]1 + \cot^2 \theta = \csc^2 \theta{.}[/latex] (Hint: Start with the identity [latex]\cos^2 \theta + \sin^2 \theta = 1[/latex] and divide both sides of the equation by [latex]\sin^2 \theta{.}[/latex])

Suppose that [latex]\cot \theta = 5[/latex] and [latex]\theta[/latex] lies in the third quadrant.

• Use the Pythagorean identity to find the value of [latex]\csc \theta{.}[/latex]
• Use identities to find the values of the other four trig functions of [latex]\theta{.}[/latex]

Suppose that [latex]\tan \theta = -2[/latex] and [latex]\theta[/latex] lies in the second quadrant.

• Use the Pythagorean identity to find the value of [latex]\sec \theta{.}[/latex]

Write each of the other five trig functions in terms of [latex]\sin t[/latex] only.

Write each of the other five trig functions in terms of [latex]\cos t[/latex] only.

Show that if the angles of a triangle are [latex]A,~B,[/latex] and [latex]C[/latex] and the opposite sides are respectively [latex]a,~b,[/latex] and [latex]c,[/latex] then

[latex]a \csc A = b \csc B = c \csc C[/latex]

Trigonometry Copyright © 2024 by Bimal Kunwor; Donna Densmore; Jared Eusea; and Yi Zhen. All Rights Reserved.

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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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Access for free at https://openstax.org/books/algebra-and-trigonometry/pages/1-introduction-to-prerequisites

• Authors: Jay Abramson
• Publisher/website: OpenStax
• 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

© Dec 8, 2021 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.