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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 𝑣𝑎𝑙𝑢𝑒+min 𝑣𝑎𝑙𝑢𝑒

2

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

Amplitude =
max 𝑣𝑎𝑙𝑢𝑒−min 𝑣𝑎𝑙𝑢𝑒

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 𝑥 𝑎𝑛𝑑 𝑦 = cos 𝑥 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 “mid”
for midline, “max” for maximum, and “min” for minimum.

a) Sine: starting with _________ at 𝑥 = 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 𝑥 = 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 𝑥 _________-> __________->________->________->________

b) 𝑦 = −2 sin 𝑥 _________-> __________->________->________->________

c) 𝑦 = − cos 𝑥 _________-> __________->________->________->________

d) 𝑦 =
1

2
cos 𝑥 _________-> __________->________->________->________

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

functions? Answer using “min”, “max” or “mid.”

Equation Starting position (when 𝑥 = 0, which is the phase shift in these cases)
𝑦 = 𝑎 sin 𝑥, 𝑎 > 0

𝑦 = 𝑎 sin 𝑥, 𝑎 0

𝑦 = 𝑎 cos 𝑥, 𝑎 > 0

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REVIEW: Transformations of Sine and Cosine
Given an equation in the form 𝑓(𝑡) = 𝐴 sin(𝐵(𝑡 − 𝐶)) + 𝑘 or 𝑓(𝑡) = 𝐴 cos(𝐵(𝑡 − 𝐶)) + 𝑘

• A is the vertical stretch, and |𝐴| is the amplitude of the function.

• B is the horizontal stretch/compression, and is related to the period, P=
2𝜋

𝐵

• k is the vertical shift and determines the midline of the function, y =k.

• 𝐶 is the phase (horizontal) shift.

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

a) 𝑦 = 3 sin(8𝑥) + 5 b) 𝑦 =
1

2
cos(3𝑥) − 2

c) 𝑦 = 3 sin (
𝜋

2
(𝑥 + 1)) − 4 d) 𝑦 = −2 cos (

3𝜋

4
𝑥) + 1

e) 𝑦 = −5 sin(2(𝑥 − 𝜋)) + 3 f) 𝑓(𝑥) = 5 − cos (4𝑥 − 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 −
( )2

➢ 𝑎2 − 𝑏2 = (𝑎 + 𝑏)(𝑎 − 𝑏)

a) 4𝑥2 − 49
= (2𝑥)2 − 72 = (2𝑥 + 7)(2𝑥 − 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

➢ 𝑎2 + 2𝑎𝑏 + 𝑏2 = (𝑎 + 𝑏)2
➢ 𝑎2 − 2𝑎𝑏 + 𝑏2 = (𝑎 − 𝑏)2

b) 𝑥2 + 6𝑥 + 9
= (𝑥)2 + 2(3)(𝑥) + 32 = (𝑥 + 3)2

c) 4𝑥2 − 20𝑥 + 25 = (2𝑥)2 − 2(2𝑥)(5) + 52
= (2𝑥 − 5)2

To factor 𝑎𝑥2 + 𝑏𝑥 + 𝑐
1. Factor 𝑎𝑥2 into 𝑎1𝑥 ∙ 𝑎2𝑥 and set up

parentheses

2. Find possible 𝑐1 𝑎𝑛𝑑 𝑐2 so that 𝑐 = 𝑐1 ∙ 𝑐2

3. Determine signs and check

d) 𝟐𝒙𝟐 − 𝒙 − 𝟑
(2𝑥 )(𝑥 )

3 = 1 ∙ 3
Different signs
Attempt #1: (2𝑥 + 1 )(𝑥 − 3)
Check: −6𝑥 + 𝑥 = −5𝑥 ≠the middle term 𝑥
Try again: Switch the numbers
Attempt #2: (2𝑥 + 3 )(𝑥 − 1)
Check: −2𝑥 + 3𝑥 = 𝑥 ≠ −𝑥 but we only
missed the sign
Try again: Change the signs
Attempt #3: (2𝑥 − 3 )(𝑥 + 1)

Check: 2𝑥 − 3𝑥 = −𝑥 √

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) 2𝑥2 + 5𝑥 − 3 = 0 b) 2𝑥2 + 𝑥 − 6 = 0

c) 3𝑥2 + 2𝑥 + 1 = 0 d) 4𝑥2 − 3 = 0

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#2 Use the unit circle to find all values of 𝜃 on the interval 0 ≤ θ < 2π that satisfies

a) 𝑦 =
1

2

b) 𝑥 = −
√3

2

c) 𝑥 = 0

d) 𝑦 = −1

#3 Find all solutions on the interval 0 ≤ θ < 2π. Give all answer in radians.

a) cos 𝑥 − 1 = 0 b) 2 sin 𝑥 + √2 = 0

REVIEW: Right triangle trigonometry Special right triangles

sin 𝜽 =
𝑶

𝑯
, tan 𝜽 =

𝑶

𝑨
, css 𝜽 =

𝑯

𝑶

cos 𝜽 =
𝑨

𝑯
, cot 𝜽 =

𝑨

𝑶
, sec 𝜽 =

𝑯

𝑨

#4 Find 𝜃 on the interval [0,
𝜋

2
)

a) tan 𝜃 =
1

√3
b) tan 𝜃 = 1 c) tan 𝜃 = 0

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REVIEW: Complete the table.

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

2𝜋

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

a) cos 𝑥 − 1 = 0 b) 2 sin 𝑥 + √2 = 0

c) 𝑡𝑎𝑛 𝑥 − 1 = 0 d) tan 𝑥 + 1 = 0

#6. Write each using only the indicated function

a) cos2 𝑥 − 2 sin 𝑥; use sine b) cos(𝑥) − 2 sin2 𝑥; use cosine

#7 Factor each expression

a) 2 sin2 𝑥 + sin 𝑥 cos 𝑥 b) 2 cos2 𝑥 + cos 𝑥 − 1

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