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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ππ’π+min π£πππ’π

2

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

Amplitude =
max π£πππ’πβmin π£ππ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 π¦ = sin π₯ πππ π¦ = cos π₯ over one period, on the interval [0, 2π]

#2. Use the graphs of the basic sine and cosine equations to answer the following in terms of βmidβ
for midline, βmaxβ for maximum, and βminβ 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) π¦ = 3 sin π₯ _________-> __________->________->________->________

b) π¦ = β2 sin π₯ _________-> __________->________->________->________

c) π¦ = β cos π₯ _________-> __________->________->________->________

d) π¦ =
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minβ, βmaxβ or βmid.β

Equation Starting position (when π₯ = 0, which is the phase shift in these cases)
π¦ = π sin π₯, π > 0

π¦ = π sin π₯, π 0

π¦ = π cos π₯, π > 0

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REVIEW: Transformations of Sine and Cosine
Given an equation in the form π(π‘) = π΄ sin(π΅(π‘ β πΆ)) + π or π(π‘) = π΄ cos(π΅(π‘ β πΆ)) + π

β’ A is the vertical stretch, and |π΄| is the amplitude of the function.

β’ B is the horizontal stretch/compression, and is related to the period, P=
2π

π΅

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

β’ πΆ is the phase (horizontal) shift.

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

a) π¦ = 3 sin(8π₯) + 5 b) π¦ =
1

2
cos(3π₯) β 2

c) π¦ = 3 sin (
π

2
(π₯ + 1)) β 4 d) π¦ = β2 cos (

3π

4
π₯) + 1

e) π¦ = β5 sin(2(π₯ β π)) + 3 f) π(π₯) = 5 β cos (4π₯ β π) 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 β
( )2

β’ π2 β π2 = (π + π)(π β π)

a) 4π₯2 β 49
= (2π₯)2 β 72 = (2π₯ + 7)(2π₯ β 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

β’ π2 + 2ππ + π2 = (π + π)2
β’ π2 β 2ππ + π2 = (π β π)2

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

c) 4π₯2 β 20π₯ + 25 = (2π₯)2 β 2(2π₯)(5) + 52
= (2π₯ β 5)2

To factor ππ₯2 + ππ₯ + π
1. Factor ππ₯2 into π1π₯ β π2π₯ and set up

parentheses

2. Find possible π1 πππ π2 so that π = π1 β π2

3. Determine signs and check

d) πππ β π β π
(2π₯ )(π₯ )

3 = 1 β 3
Different signs
Attempt #1: (2π₯ + 1 )(π₯ β 3)
Check: β6π₯ + π₯ = β5π₯ β the middle term π₯
Try again: Switch the numbers
Attempt #2: (2π₯ + 3 )(π₯ β 1)
Check: β2π₯ + 3π₯ = π₯ β  βπ₯ but we only
missed the sign
Try again: Change the signs
Attempt #3: (2π₯ β 3 )(π₯ + 1)

Check: 2π₯ β 3π₯ = βπ₯ β

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

#1 Solve the following problems.

a) 2π₯2 + 5π₯ β 3 = 0 b) 2π₯2 + π₯ β 6 = 0

c) 3π₯2 + 2π₯ + 1 = 0 d) 4π₯2 β 3 = 0

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#2 Use the unit circle to find all values of π on the interval 0 β€ ΞΈ < 2Ο that satisfies

a) π¦ =
1

2

b) π₯ = β
β3

2

c) π₯ = 0

d) π¦ = β1

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

a) cos π₯ β 1 = 0 b) 2 sin π₯ + β2 = 0

REVIEW: Right triangle trigonometry Special right triangles

sin π½ =
πΆ

π―
, tan π½ =

πΆ

π¨
, css π½ =

π―

πΆ

cos π½ =
π¨

π―
, cot π½ =

π¨

πΆ
, sec π½ =

π―

π¨

#4 Find π on the interval [0,
π

2
)

a) tan π =
1

β3
b) tan π = 1 c) tan π = 0

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REVIEW: Complete the table.

π¬π’π§(π) ππ¨π¬(π) π­ππ§ (π) ππ¨π­ (π) π¬ππ (π) ππ¬π (π)
Period

2π

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

a) cos π₯ β 1 = 0 b) 2 sin π₯ + β2 = 0

c) π‘ππ π₯ β 1 = 0 d) tan π₯ + 1 = 0

#6. Write each using only the indicated function

a) cos2 π₯ β 2 sin π₯; use sine b) cos(π₯) β 2 sin2 π₯; use cosine

#7 Factor each expression

a) 2 sin2 π₯ + sin π₯ cos π₯ b) 2 cos2 π₯ + cos π₯ β 1

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