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Trigonometry
The Six Trigonometric Functions

1
Angles, Degrees, and Special Triangles

Problem 1

Welcome to Trigonometry from the author of your textbook.

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Mr. McKeague cc
Mr. McKeague
Problem 2

Give the complement and supplement of each angle.

  1. \(40^{\circ}\)

  2. \(110^{\circ}\)

  3. \(\theta\)

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Mr. McKeague
Stefanie cc
Stefanie
Gordon cc
Gordon
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Gordon
Problem 3

The hypotenuse of a right triangle is five. If the two legs are given by two consecutive integers, find the length of the two legs.

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Mr. McKeague cc
Mr. McKeague
Julieta cc
Julieta
Julieta cc spanish language icon
Julieta
Problem 4

The vertical rise of the Forest Double chair lift is \(1,170\) feet and the length of the chair lift is \(5,750\) feet. To the nearest foot, find the horizontal distance covered by a person riding in the lift.

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Stefanie cc
Stefanie
Aaron cc
Aaron
Edwin cc spanish language icon
Edwin
Problem 5

If the shortest side of a \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle is \(5\), find the other two sides.

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Mr. McKeague cc
Mr. McKeague
Stefanie cc
Stefanie
Gordon cc
Gordon
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Gordon
Problem 6

A ladder is leaning against a wall. The top of the ladder is \(4\) feet above the ground and the bottom of the ladder makes an angle of \(60^{\circ}\) with the ground. How long is the ladder, and how far from the wall is the bottom of the ladder?

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Mr. Perez cc
Mr. Perez
Stefanie cc
Stefanie
Gordon cc
Gordon
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Gordon
Problem 7

A \(10\)-foot rope connects the top of a tent pole to the ground. If the rope makes an angle of \(45^{\circ}\) with the ground, find the length of the tent pole.

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Mr. Perez cc
Mr. Perez
Stefanie cc
Stefanie
Gordon cc
Gordon
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Gordon

2
The Rectangular Coordinate System

Problem 1

Graph the ordered pairs \((-1,3), (2,5), (0,0), (0,-3),\) and \((4,0)\).

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Molly S. cc
Molly S.
Preston cc
Preston
Betsy cc
Betsy
David cc spanish language icon
David
Problem 2

Graph \(y=-\displaystyle\frac{1}{3}x\)

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Mr. McKeague cc
Mr. McKeague
Betsy cc
Betsy
Preston cc
Preston
Edwin spanish language icon
Edwin
Problem 3

Find the \(x\)- and \(y\)-intercepts for \(3x-2y=6\), and graph.

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Betsy cc
Betsy
Preston cc
Preston
David cc spanish language icon
David
Problem 4

Graph the equation \(y=x^2\).

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Stefanie cc
Stefanie
Betsy cc
Betsy
Preston cc
Preston
Edwin cc spanish language icon
Edwin
Problem 5

Graph: \(y=x^2-3\)

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Mr. McKeague cc
Mr. McKeague
Betsy cc
Betsy
Preston cc
Preston
Julieta cc spanish language icon
Julieta
Problem 6

As a human cannonball, David Smith Jr., reached a height of \(70\) feet before landing in a net \(160\) feet from the cannon. Sketch the graph of his path, and then find the equation of the graph.

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Mr. McKeague cc
Mr. McKeague
Betsy cc
Betsy
CJ cc
CJ
Julieta cc spanish language icon
Julieta
Problem 7

Sketch \(x^2+y^2=9\)

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Stefanie cc
Stefanie
Preston cc
Preston
Edwin cc spanish language icon
Edwin
Problem 8

If \((x-1)^2 +(y+3)^2=4\) find the center, radius, and graph the circle.

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Mr. McKeague cc
Mr. McKeague
Betsy cc
Betsy
Gordon cc
Gordon
Gordon spanish language icon
Gordon
Problem 9

Suppose a Ferris wheel has a radius of 50 feet, with the bottom of the wheel 10 feet above the ground. Let t represent the time (in minutes). If the wheel completes one revolution in 2 minutes, find the distance a rider is above the ground when the ride starts (t = 0), after 30 seconds (t =  1/ 2 ), at one minute (t = 1), 30 seconds later (t =  3 / 2 ), and at 2 minutes (t = 2).

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Mr. McKeague cc
Mr. McKeague
Julieta cc
Julieta
Julieta cc spanish language icon
Julieta
Problem 10

Find the distance between the points \((-1,5)\) and \((2,1)\).

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Betsy cc
Betsy
CJ cc
CJ
Julieta cc spanish language icon
Julieta
Problem 11

Find the distance from the origin to the point \((x,y)\).

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Betsy cc
Betsy
CJ cc
CJ
Julieta cc spanish language icon
Julieta
Problem 12

Draw an angle of \(45^{\circ}\) in standard position and find a point on the terminal side.

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Stefanie cc
Stefanie
CJ cc
CJ
Julieta cc spanish language icon
Julieta
Problem 13

Draw \(-90^{\circ}\) in standard position and find the two positive angles and two negative angles that are coterminal with \(-90^{\circ}\).

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Stefanie cc
Stefanie
CJ cc
CJ
Julieta cc spanish language icon
Julieta

3
Definition I: Trigonometric Functions

Problem 1

Welcome from Mr. McKeague

In this section you will see the definition for the trigonometric functions for the first time. Notice that Mr. McKeague emphasizes that the definition must be memorized.

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Mr. McKeague cc
Mr. McKeague
Problem 2

Find the six trigonometric functions of \(\theta\) if \(\theta\) is in standard position and the point \((-2,3)\) is on the terminal side of \(\theta\).

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Mr. McKeague cc
Mr. McKeague
Stefanie cc
Stefanie
Gordon cc
Gordon
Gordon cc spanish language icon
Gordon
Problem 3

Find the sine and cosine of \(45^{\circ}\).

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Mr. McKeague cc
Mr. McKeague
Betsy cc
Betsy
Aaron cc
Aaron
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Edwin
Problem 4

Find the six trigonometric functions of \(270^{\circ}\).

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Mr. McKeague cc
Mr. McKeague
Betsy cc
Betsy
Aaron cc
Aaron
Julieta cc spanish language icon
Julieta
Problem 5

If \(\sin{\theta}=-\displaystyle\frac{5}{13}\), and \(\theta\) terminates in quadrant III, find \(\cos{\theta}\) and \(\tan{\theta}\).

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Mr. McKeague
Stefanie cc
Stefanie
Gordon cc
Gordon
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Gordon
Problem 6

Suppose a Ferris wheel has a radius of 50 feet, with the bottom of the wheel 10 feet above the ground. If the wheel completes one revolution in 2 minutes, find the distance a rider is above the ground when the rider is at point A on the wheel.

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Mr. McKeague cc
Mr. McKeague
Julieta cc
Julieta
Julieta spanish language icon
Julieta
Problem 7

Mini Lecture

  1. Find the six trigonometric functions of \(\theta\) if \((-5,12)\) is on the terminal side.

  2. Find sine, cosine, and tangent of \(-45^{\circ}\).

  3. If \(\tan{\theta}=\displaystyle\frac{3}{4}\) and \(\theta\in\) QI, find \(\sin{\theta}\) and \(\cos{\theta}\).

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Mr. McKeague cc
Mr. McKeague

4
Introduction to Identities

Problem 1

Welcome from Mr. McKeague and Edwin

Just to be sure you understand the importance of the definition for the trigonometric functions, and that it must be memorized, here it is again from Mr. McKeague and Edwin.

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Mr. McKeague cc
Mr. McKeague
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Edwin
Problem 2

If \(\sin{\theta}=\displaystyle\frac{3}{5}\), then \(\csc{\theta}=\)?

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Samantha cc
Samantha
Stefanie cc
Stefanie
Gordon cc
Gordon
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Gordon
Problem 3

If \(\cos{\theta}=-\displaystyle\frac{\sqrt{3}}{2}\), then \(\sec{\theta}=\)?

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Samantha cc
Samantha
Betsy cc
Betsy
CJ cc
CJ
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Cynthia
Problem 4

If \(\tan{\theta}=2\), then \(\cot{\theta}=\)?

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Mr. McKeague
Stefanie cc
Stefanie
Gordon cc
Gordon
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Gordon
Problem 5

If \(\csc{\theta}=a\) then \(\sin{\theta}=\)?

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Betsy
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CJ
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Cynthia
Problem 6

If \(\sec{\theta}=1\), then \(\cos{\theta}=\)?

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Stefanie cc
Stefanie
Gordon cc
Gordon
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Gordon
Problem 7

If \(\cot{\theta}=-1\), then \(\tan{\theta}=\)?

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Betsy cc
Betsy
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CJ
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Cynthia
Problem 8

If \(\sin{\theta}=-\displaystyle\frac{3}{5}\) and \(\cos{\theta}=\displaystyle\frac{4}{5}\), find \(\tan{\theta}\) and \(\cot{\theta}\).

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Stefanie cc
Stefanie
Gordon cc
Gordon
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Gordon
Problem 9

If \(\sin{\theta}=\displaystyle\frac{3}{5}\), then \(\sin^2{\theta}=\)?

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Stefanie cc
Stefanie
CJ cc
CJ
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Julieta
Problem 10

If \(\cos{\theta}=-\displaystyle\frac{1}{2}\), then \(\cos^3{\theta}=\)?

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Gordon cc
Gordon
Stefanie cc
Stefanie
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Gordon
Problem 11

If \(\sin{\theta}=\displaystyle\frac{3}{5}\) and \(\theta\) terminates in QII, find \(\cos{\theta}\) and \(\tan{\theta}\).

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Stefanie cc
Stefanie
CJ cc
CJ
Julieta cc spanish language icon
Julieta
Problem 12

If \(\cos{\theta}=\displaystyle\frac{1}{2}\) and \(\theta\) terminates in QIV, find the remaining trigonometric ratios for \(\theta\).

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Stefanie cc
Stefanie
Gordon cc
Gordon
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Gordon
Problem 13

Mini Lecture

  1. If \(\sin{\theta}=\frac{4}{5}\) and \(\csc{\theta}\).

  2. Find \(\cot{\theta}\) if \(\sin{\theta}=-\frac{5}{13}\) and \(\cos{\theta}=-\frac{12}{13}\).

  3. Find \(\cos{\theta}\) if \(\sin{\theta}=\frac{1}{3}\) and \(\theta\in\) QII

  4. \(\cos{\theta}=\frac{12}{13}\) with \(\theta\in\) QI, then find \(\sin{\theta}\), \(\sec\theta\), \(\tan\theta\), \(\cot\theta\), and \(\csc\theta\).

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Mr. McKeague cc
Mr. McKeague

5
More on Identities

Problem 1

Write \(\tan\theta\) in terms of \(\sin\theta\).

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Stefanie
Gordon cc
Gordon
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Gordon
Problem 2

Write \(\sec\theta\tan\theta\) in terms of \(\sin\theta\) and \(\cos\theta\), and then simplify.

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Betsy cc
Betsy
CJ cc
CJ
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Cynthia
Problem 3

Add \(\displaystyle\frac{1}{\sin\theta}+\displaystyle\frac{1}{\cos\theta}\).

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Stefanie cc
Stefanie
CJ cc
CJ
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Cynthia
Problem 4

Multiply \((\sin\theta +2)(\sin\theta -5)\).

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Betsy cc
Betsy
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CJ
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Cynthia
Problem 5

Show that the following statement is true by transforming the left side into the right side. \[\cos\theta\tan\theta=\sin\theta\]

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Stefanie cc
Stefanie
CJ cc
CJ
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Julieta
Problem 6

Prove the identity \((\sin\theta + \cos\theta)^2=1+2\sin\theta\cos\theta\)

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CJ cc
CJ
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Julieta
Problem 7

Mini Lecture

  1. Write \(\tan\theta\) in terms of \(\cos\theta\) only.

  2. Add: \(\sin\theta+\displaystyle\frac{1}{\cos\theta}\)

  3. Prove \(\cos\theta\tan\theta=\sin\theta\)

  4. Prove \(\displaystyle\frac{\sin\theta}{\csc\theta}=\sin^2\theta\)

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Mr. McKeague cc
Mr. McKeague

6
Spotlight on Julieta

Problem 1

Spotlight on Julieta

Julieta speaks about her experience with A.V.I.D.

Achievement Via Individual Determination is a non-profit organization that provides professional learning for educators to improve college readiness for all students, especially those traditionally underrepresented in higher education.

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Julieta cc
Julieta