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Math 1083 Worksheet 12 Getting Ready for Modeling with Trigonometric Functions

Objectives:

1. Identify key features from sinusoidal curves

2. Five key points for the basic sine and cosine graphs

3. Determine the key features of sinusoidal curves from equations

A midline is a horizontal line that divides the graph in half vertically.

Midline Equation for sine and cosine: y =
max π‘£π‘Žπ‘™π‘’π‘’+min π‘£π‘Žπ‘™π‘’π‘’

2

Amplitude is the distance from the midline to the maximum or minimum.

Amplitude =
max π‘£π‘Žπ‘™π‘’π‘’βˆ’min π‘£π‘Žπ‘™π‘’π‘’

2

#1 For each given graph, find the minimum, maximum, amplitude and the midline equation.

a) b)

maximum: ___________ minimum: _________ maximum: ___________ minimum: _________

amplitude: _______ Midline: _______________ amplitude: _______ Midline: _______________

c) d)

maximum: ___________ minimum: _________ maximum: ___________ minimum: _________

amplitude: _______ Midline: _______________ amplitude: _______ Midline: _______________

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Review: the graphs of 𝑦 = sin π‘₯ π‘Žπ‘›π‘‘ 𝑦 = cos π‘₯ over one period, on the interval [0, 2πœ‹]

#2. Use the graphs of the basic sine and cosine equations to answer the following in terms of β€œmid”
for midline, β€œmax” for maximum, and β€œmin” for minimum.

a) Sine: starting with _________ at π‘₯ = 0

b) Find the height of the five points that correspond to quadrantal points [the first three are

done]

___mid___-> __max_____->__mid____->________->________

c) Cosine: starting with __________ at π‘₯ = 0

d) Find the height of the five points that correspond to quadrantal points

_________-> __________->________->________->________

#3 For each equation below, what is the pattern of the five key points?

a) 𝑦 = 3 sin π‘₯ _________-> __________->________->________->________

b) 𝑦 = βˆ’2 sin π‘₯ _________-> __________->________->________->________

c) 𝑦 = βˆ’ cos π‘₯ _________-> __________->________->________->________

d) 𝑦 =
1

2
cos π‘₯ _________-> __________->________->________->________

e) What can you summarize about the starting position (when x = 0) of the sine and cosine

functions? Answer using β€œmin”, β€œmax” or β€œmid.”

Equation Starting position (when π‘₯ = 0, which is the phase shift in these cases)
𝑦 = π‘Ž sin π‘₯, π‘Ž > 0

𝑦 = π‘Ž sin π‘₯, π‘Ž 0

𝑦 = π‘Ž cos π‘₯, π‘Ž > 0

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REVIEW: Transformations of Sine and Cosine
Given an equation in the form 𝑓(𝑑) = 𝐴 sin(𝐡(𝑑 βˆ’ 𝐢)) + π‘˜ or 𝑓(𝑑) = 𝐴 cos(𝐡(𝑑 βˆ’ 𝐢)) + π‘˜

β€’ A is the vertical stretch, and |𝐴| is the amplitude of the function.

β€’ B is the horizontal stretch/compression, and is related to the period, P=
2πœ‹

𝐡

β€’ k is the vertical shift and determines the midline of the function, y =k.

β€’ 𝐢 is the phase (horizontal) shift.

#4 For each of the following equations, find the amplitude, period, phase shift and midline.

a) 𝑦 = 3 sin(8π‘₯) + 5 b) 𝑦 =
1

2
cos(3π‘₯) βˆ’ 2

c) 𝑦 = 3 sin (
πœ‹

2
(π‘₯ + 1)) βˆ’ 4 d) 𝑦 = βˆ’2 cos (

3πœ‹

4
π‘₯) + 1

e) 𝑦 = βˆ’5 sin(2(π‘₯ βˆ’ πœ‹)) + 3 f) 𝑓(π‘₯) = 5 βˆ’ cos (4π‘₯ βˆ’ πœ‹) 1

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Math 1083 Worksheet 11 Getting Ready for Solving Trigonometric Equations

Objectives:
3. Solve equations by factoring

4. Review the unit circle

5. Review right triangle trigonometry

(1) Review Solving Equations by Factoring

REVIEW: Special Identities
Difference of squares: in the form ( )2 βˆ’
( )2

➒ π‘Ž2 βˆ’ 𝑏2 = (π‘Ž + 𝑏)(π‘Ž βˆ’ 𝑏)

a) 4π‘₯2 βˆ’ 49
= (2π‘₯)2 βˆ’ 72 = (2π‘₯ + 7)(2π‘₯ βˆ’ 7)

Perfect square trinomials: First term and
last terms are perfect squares and the middle
term is twice the product of first and last
terms

➒ π‘Ž2 + 2π‘Žπ‘ + 𝑏2 = (π‘Ž + 𝑏)2
➒ π‘Ž2 βˆ’ 2π‘Žπ‘ + 𝑏2 = (π‘Ž βˆ’ 𝑏)2

b) π‘₯2 + 6π‘₯ + 9
= (π‘₯)2 + 2(3)(π‘₯) + 32 = (π‘₯ + 3)2

c) 4π‘₯2 βˆ’ 20π‘₯ + 25 = (2π‘₯)2 βˆ’ 2(2π‘₯)(5) + 52
= (2π‘₯ βˆ’ 5)2

To factor π‘Žπ‘₯2 + 𝑏π‘₯ + 𝑐
1. Factor π‘Žπ‘₯2 into π‘Ž1π‘₯ βˆ™ π‘Ž2π‘₯ and set up

parentheses

2. Find possible 𝑐1 π‘Žπ‘›π‘‘ 𝑐2 so that 𝑐 = 𝑐1 βˆ™ 𝑐2

3. Determine signs and check

d) πŸπ’™πŸ βˆ’ 𝒙 βˆ’ πŸ‘
(2π‘₯ )(π‘₯ )

3 = 1 βˆ™ 3
Different signs
Attempt #1: (2π‘₯ + 1 )(π‘₯ βˆ’ 3)
Check: βˆ’6π‘₯ + π‘₯ = βˆ’5π‘₯ β‰ the middle term π‘₯
Try again: Switch the numbers
Attempt #2: (2π‘₯ + 3 )(π‘₯ βˆ’ 1)
Check: βˆ’2π‘₯ + 3π‘₯ = π‘₯ β‰  βˆ’π‘₯ but we only
missed the sign
Try again: Change the signs
Attempt #3: (2π‘₯ βˆ’ 3 )(π‘₯ + 1)

Check: 2π‘₯ βˆ’ 3π‘₯ = βˆ’π‘₯ √

Note: You may open the link https://ggbm.at/uyFoojWM. Use the applet to review how to factor a

quadratic expression.

#1 Solve the following problems.

a) 2π‘₯2 + 5π‘₯ βˆ’ 3 = 0 b) 2π‘₯2 + π‘₯ βˆ’ 6 = 0

c) 3π‘₯2 + 2π‘₯ + 1 = 0 d) 4π‘₯2 βˆ’ 3 = 0

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#2 Use the unit circle to find all values of πœƒ on the interval 0 ≀ ΞΈ < 2Ο€ that satisfies

a) 𝑦 =
1

2

b) π‘₯ = βˆ’
√3

2

c) π‘₯ = 0

d) 𝑦 = βˆ’1

#3 Find all solutions on the interval 0 ≀ ΞΈ < 2Ο€. Give all answer in radians.

a) cos π‘₯ βˆ’ 1 = 0 b) 2 sin π‘₯ + √2 = 0

REVIEW: Right triangle trigonometry Special right triangles

sin 𝜽 =
𝑢

𝑯
, tan 𝜽 =

𝑢

𝑨
, css 𝜽 =

𝑯

𝑢

cos 𝜽 =
𝑨

𝑯
, cot 𝜽 =

𝑨

𝑢
, sec 𝜽 =

𝑯

𝑨

#4 Find πœƒ on the interval [0,
πœ‹

2
)

a) tan πœƒ =
1

√3
b) tan πœƒ = 1 c) tan πœƒ = 0

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REVIEW: Complete the table.

𝐬𝐒𝐧(𝒙) 𝐜𝐨𝐬(𝒙) 𝐭𝐚𝐧 (𝒙) 𝐜𝐨𝐭 (𝒙) 𝐬𝐞𝐜 (𝒙) 𝐜𝐬𝐜 (𝒙)
Period

2πœ‹

#5 Find all solutions. Give the general answer in radians.

a) cos π‘₯ βˆ’ 1 = 0 b) 2 sin π‘₯ + √2 = 0

c) π‘‘π‘Žπ‘› π‘₯ βˆ’ 1 = 0 d) tan π‘₯ + 1 = 0

#6. Write each using only the indicated function

a) cos2 π‘₯ βˆ’ 2 sin π‘₯; use sine b) cos(π‘₯) βˆ’ 2 sin2 π‘₯; use cosine

#7 Factor each expression

a) 2 sin2 π‘₯ + sin π‘₯ cos π‘₯ b) 2 cos2 π‘₯ + cos π‘₯ βˆ’ 1

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