Grade 12 Pre-Calculus Mathematics

Notebook

Chapter 5

Trigonometric Functions and

Graphs

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Chapter 5 – Homework

Section Page Questions

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Chapter 5: TRIGONOMETRIC FUNCTIONS AND GRAPHS

5.1/5.3 – Graphing the Sine, Cosine and Tangent Functions

Periodic Functions: A function that repeats itself over regular intervals (cycles) of its domain.

Period: The horizontal length of the interval over which a function travels one full rotation or cycle. (The distance from one point to that exact same point again.)

Amplitude: The vertical distance from the horizontal central axis the maximum or the minimum. Note that the central axis is not always the x-axis.

Sinusoidal Functions: Oscillating periodic functions, like siny x= or cosy x= .

Complete a table of values for y = sin

θ 0

2

2

3

2

y

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Complete a table of values for y = cos

θ 0

2

2

3

2

y

Complete a table of values for y = tan

θ 0 2

2

3

2

y

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Complete the following chart for each of the functions below.

y = sin y = cos y = tan

Amplitude

Period

Domain

Range

Zeroes (x-intercepts)

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Introduction to Transformations We can apply what we learned in our transformations unit to the sinusoidal functions

Let’s explore how the values of a, b, c and d affect the characteristics below. Complete the chart below.

amplitude range period

1) sin2=y

2) 1sin += y

3) )cos( −=y

4) )2cos( =y

5) 4cos3 −= y

6) 3cos−=y

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Chapter 5: TRIGONOMETRIC FUNCTIONS AND GRAPHS

5.2 – Transformations of Sinusoidal Functions (Part 1)

Before graphing sinusoidal functions, they must be in the form

( )( )siny a b x c d= − +

Example 1

Sketch the graph of 2

3sin 2 2.3

y x

= − +

1. Re-write the equation in factored form:

2. Recall that the graph of siny x= looks like:

3. Determine the values of a, b, c, and d.

a = b = c = d =

x

y

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4. Sketch! (a few more steps to go)

3sin 2 23

y x

= − +

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Example 2

( ) 212

sin3 −

−= xy

Example 3

34

2cos4 +

+−=

xy

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5.2 Part 1 Homework Provide a graph of each of the following functions. Include at least one full period of the function

1)

2)

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3)

4)

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5)

Challenge Write the equation of the following graph in both the form

or

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Chapter 5: TRIGONOMETRIC FUNCTIONS AND GRAPHS

5.2 – Transformations of Sinusoidal Functions (Part 2)

A sinusoidal function is expressed in the form:

dcxbay +−= ))(sin( or dcxbay +−= ))(cos(

Properties of the graph

a

d

b

c = sine funcion

c = cosine function

Example 1: Write the equation of the following graphs in the form dcxbay +−= ))(sin( or dcxbay +−= ))(cos(

a)

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b)

c) The point (–2,7) is a maximum of a sinusoidal function.

The next consecutive maximum occurs at the point (6, 7). The range of the function is [–1,7].

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5.2 Part 2: Writing Equations Homework

Write an equation for each of the following in terms of y = asin(b(x - c)) + d

1) 2)

3)

x- 2 - 2

y

- 4

- 2

2

4

x- 2 - 2

y

- 4

- 2

2

4

x- 2 - 2

y

- 4

- 2

2

4

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Write an equation for each of the following in terms of y = acos(b(x – c)) + d 4) 5)

6)

x- 2 - 2

y

- 4

- 2

2

4

x- 2 - 2

y

- 4

- 2

2

4

x- 4 - 2 2 4

y

- 4

- 2

2

4

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Chapter 5: TRIGONOMETRIC FUNCTIONS AND GRAPHS

5.4 – Equations of Graphs and Functions (Part 1)

Example 1

Graphically solve the equation 3

2sin −= over the interval 20 x .

Note: You can verify your solution by solving the equation algebraically.

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Chapter 5: TRIGONOMETRIC FUNCTIONS AND GRAPHS

5.4 – Equations of Graphs and Functions (Part 2) Example 1

At a seaport the depth of the water, h meters, at time, t hours, during a certain day is given by this formula.

( 4.00)

( ) 1.8sin 2 3.112.4

th t

− = +

a) State the:

i) period ii) amplitude iii) phase shift

b) What is the maximum depth of the water? When does it occur? c) What is the depth of the water at 5:00 am?

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Example 2

A tsunami is a very fast-moving ocean wave caused by earthquakes that occur underwater. The water will first move down from its normal level, then move an equal distance above the normal level, then finally back to the normal level. The period of a tsunami is 16 minutes with amplitude of 8 meters. The normal depth of water at Crescent Beach is 6 meters. a) What is the maximum and minimum height of water caused by the tsunami at Crescent

Beach? b) Write a sinusoidal function to represent the tsunami when it first reaches Crescent Beach.

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5.4 Part 2: Homework Assignment

Answer the following questions on a separate piece of paper. 1) The pedals on a bike have a maximum height of 30cm above the ground and a minimum

distance of 8cm above the ground. A person pedals at a constant rate of 20 cycles per minute. (Assume that you begin pedalling with your foot at the maximum height that the pedal reaches)

a) What is the period in seconds for this periodic function? b) Determine an equation for this periodic function.

2) A ferris wheel of radius 25 meters, placed one meter above the ground varies sinusoidally

with time. The ferris wheel makes one rotation every 24 seconds, with a person sitting 26 meters from the ground and rising when it starts to rotate.

a) Write a sinusoidal function that describes the function from a person’s starting

point. b) How high above the ground would a person be 16 seconds after the ferris wheel

starts moving? 3) Tides are a periodic rise and fall of water in the ocean. A low tide of 4.2 meters in White

Rock, B.C. occurs at 4:30 am, and the next high tide of 11.8 meters occurs at 11:30 am the same day.

a) Write a sinusoidal function that describes the tide flow. b) What is the tide height at 1:15 pm that same day?

4) The depth of water in a harbour is given by the equation ( ) 4.5cos(0.16 ) 13.7d t t= − + ,

where d(t) is the depth, in meters, and t is the time, in hours, after low tide.

Sketch the graph of d(t)

5) A sinusoidal function has a maximum at ,53

and the next nearest minimum is at

5, 3

3

−

. Write an equation to represent this graph.

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