Graph sine, cosine, and tangent, and model with sinusoidal functions

Alberta Grade 12 Mathematics

Trigonometry

Sine, cosine, tangent, and sinusoidal models

A repeating situation can often be described with a trigonometric graph. For example, the height of a seat on a turning ride rises and falls in a regular pattern. In this lesson, you will review the angle and coordinate ideas needed for trig graphs, learn their key features, and use sine or cosine to model a repeating situation. You will also compare the parent tangent graph with sine and cosine.

What you will learn

Identify the key features of sine, cosine, and tangent graphs.
Graph sine and cosine using amplitude, period, and vertical shift.
Build and interpret a sinusoidal model from a repeating situation.

1. Prerequisites: angles and coordinates

An angle can be measured in degrees or radians. A full turn is

360^\circ

or

2\pi

radians. In many graphing questions, the horizontal input is an angle measured in radians. Check the units before using a period or reading a graph.

On the unit circle, a point at angle

x

has coordinates

(\cos x,\sin x)

. This means cosine gives the horizontal coordinate and sine gives the vertical coordinate. You can use familiar exact values, such as

\sin 0=0

,

\cos 0=1

, and

\sin(\pi/2)=1

.

The tangent ratio is

\tan x=\sin x/\cos x

, when

\cos x\ne 0

. Tangent is undefined when its denominator is zero. These inputs become vertical asymptotes on the tangent graph.

\tan x=\frac{\sin x}{\cos x}

One full turn is $2\pi$ radians.
Sine and cosine are defined for every real input.
Tangent is undefined when cosine is zero.

2. Read the parent graphs

The parent graphs are the graphs with no shifts or stretches:

y=\sin x

,

y=\cos x

, and

y=\tan x

. A cycle is one complete repeating pattern. Sine and cosine each have period

2\pi

, meaning their values repeat every

2\pi

units along the horizontal axis.

Both sine and cosine stay between

-1

and

1

. The sine graph begins at zero, rises to one, returns to zero, falls to negative one, and returns to zero over one cycle. The cosine graph begins at one, falls to negative one, and returns to one over one cycle.

The tangent graph has period

\pi

. It crosses the horizontal axis at integer multiples of

\pi

. It is undefined at

x=\pi/2+k\pi

, where

k

is any integer. Each such input is a vertical asymptote: the graph approaches that vertical line but does not include points on it. Tangent is not bounded above or below.

\sin(x+2\pi)=\sin x,\quad \cos(x+2\pi)=\cos x,\quad \tan(x+\pi)=\tan x

Sine and cosine have period $2\pi$ and range $[-1,1]$ .
Tangent has period $\pi$ and is undefined at $x=\pi/2+k\pi$ .
A graph's range is the set of possible output values.

3. Transform and model sine and cosine

A sinusoidal function is a shifted or stretched sine or cosine function. A useful form is

y=a\sin(b(x-c))+d

or

y=a\cos(b(x-c))+d

. The value

a

controls the vertical stretch and reflection; the amplitude is |a|. The value

d

shifts the graph up or down. The line

y=d

is the midline.

The value

b

changes the period. The period is

2\pi/|b|

, provided

b\ne0

. The value

c

shifts the graph horizontally. With this form, the cycle begins at input

c

; for a sine graph it begins on the midline, and for a cosine graph it begins at a maximum if

a>0

.

To sketch one cycle, mark the start, maximum, midline crossing, minimum, and ending point. These five points are separated by one quarter of the period. Join them with a smooth wave. If

a

is negative, the graph is reflected across its midline.

For a repeating context, first identify the maximum and minimum. The amplitude is half their difference, and the midline is their average. Find the period from the time between matching points, such as consecutive maxima. Then choose sine or cosine based on the starting behaviour. Include units and state a reasonable domain for the situation.

y=a\sin(b(x-c))+d,\quad P=\frac{2\pi}{|b|}

Amplitude: |a|; midline: $y=d$ .
Period: $2\pi/|b|$ .
For a model, connect each parameter to a measured feature and include units.

4. A reliable graphing process

For a sine or cosine graph, identify

a

,

b

,

c

, and

d

before plotting. Calculate the amplitude, midline, and period. Divide the period by four to find the spacing between the five key points in one cycle.

For tangent, mark its zeros and vertical asymptotes before drawing. The parent graph has zeros at

x=k\pi

and vertical asymptotes at

x=\pi/2+k\pi

. Draw the increasing branch between consecutive asymptotes. Do not assign a value to the function at an asymptote.

Label axes with the correct units.
Use exact values when they are known; label decimal readings as approximations.
A graph window shows only part of a graph, not necessarily its full domain.

Worked example

Graph a transformed cosine function

Describe and sketch one cycle of

y=2\cos(2(x-\pi/4))-1

.

One cycle of the cosine function

The plotted interval is one cycle of the full periodic graph.

Find the graph features
The coefficient of cosine is $2$ , so the amplitude is $2$ . The vertical shift is $-1$ , so the midline is $y=-1$ . Since $b=2$ , the period is $2\pi/2=\pi$ . The horizontal shift is $\pi/4$ to the right.
Mark five key points
One cycle starts at $x=\pi/4$ . The quarter-period is $\pi/4$ . A positive cosine starts at its maximum, which is $-1+2=1$ . Continue by moving down to the midline, to the minimum, back to the midline, and to the maximum.
$(\pi/4,1),\ (\pi/2,-1),\ (3\pi/4,-3),\ (\pi,-1),\ (5\pi/4,1)$

Answer: Amplitude

2

, midline

y=-1

, period

\pi

, and one-cycle domain

[\pi/4,5\pi/4]

. The range is

[-3,1]

.

Check: The first and last points are both maxima, and their horizontal separation is

5\pi/4-\pi/4=\pi

, the period.

Worked example

Build a model for a repeating height

A platform moves up and down. Its height ranges from 1.2 m to 5.2 m. It reaches a maximum at time 0 s and again at 8 s. Write a cosine model for its height

h

in metres at time

t

in seconds, for

0\le t\le 8

.

Platform height over one cycle

The window shows the model from 0 to 8 seconds, not its full domain.

Find amplitude and midline
The amplitude is half the difference between the maximum and minimum. The midline is their average. These values describe the size and central height of the motion.
$A=\frac{5.2-1.2}{2}=2,\qquad d=\frac{5.2+1.2}{2}=3.2$
Find the period and write the model
The time between consecutive maximum heights is one period, so $P=8$ seconds. In the form $h=A\cos(bt)+d$ , use $b=2\pi/P=\pi/4$ . A cosine model with positive amplitude starts at a maximum, matching the stated height at time zero.
$h=2\cos\left(\frac{\pi}{4}t\right)+3.2$

Answer: A suitable model is

h=2\cos(\pi t/4)+3.2

, for

0\le t\le8

. Height is in metres and time is in seconds.

Check: At

t=0

, the model gives

5.2

m. At

t=4

, it gives

1.2

m, and at

t=8

, it gives

5.2

m.

Worked example

Read the parent tangent graph

For

y=\tan x

, identify the zeros and vertical asymptotes in the interval

[-\pi,\pi]

, and state the domain restrictions there.

Parent tangent graph

This is a window of the full tangent graph. Vertical lines mark excluded inputs.

Find the zeros
Tangent is zero when sine is zero and cosine is nonzero. In the stated interval, the zeros occur at the integer multiples of $\pi$ .
$x=-\pi,\ 0,\ \pi$
Find the excluded inputs
Tangent is undefined when cosine is zero. In this interval, those inputs are $-\pi/2$ and $\pi/2$ . They are vertical asymptotes, so they must be excluded from the domain.
$x\ne-\frac{\pi}{2},\quad x\ne\frac{\pi}{2}$

Answer: Zeros:

-\pi,0,\pi

. Vertical asymptotes:

-\pi/2

and

\pi/2

. On the stated interval, the domain excludes those two inputs.

Check: At each listed asymptote,

\cos x=0

, so the quotient defining tangent has a zero denominator.

Common mistakes and how to avoid them

Using

2\pi b

as the period of a sine or cosine graph.

Correction: For

a\sin(b(x-c))+d

or

a\cos(b(x-c))+d

, use

2\pi/|b|

.

Calling the vertical shift the amplitude.

Correction: Amplitude is |a|. The vertical shift is

d

, and it sets the midline.

Including an input where tangent is undefined.

Correction: Exclude every input where cosine is zero. Show these locations as vertical asymptotes.

Using the maximum height as the amplitude in a model.

Correction: Find amplitude as half the difference between the maximum and minimum.

Lesson summary

Sine and cosine have period $2\pi$ and range $[-1,1]$ before transformations.
For a sinusoidal model, use amplitude |a|, midline $y=d$ , and period $2\pi/|b|$ .
Tangent has period $\pi$ and is undefined when cosine is zero.
Check units, domain, and whether your model starts at a maximum, minimum, or midline.

Check your understanding

Question 1

What is the period of

y=3\sin(4x)+2

?

$8\pi$
$\pi/2$
$4\pi$
$2\pi$

Show answer and explanation

\pi/2

The period is

2\pi/|4|=\pi/2

.

Question 2

A sinusoidal quantity has maximum 14 and minimum 6. What are its amplitude and midline?

Amplitude 14; midline $y=6$
Amplitude 8; midline $y=10$
Amplitude 4; midline $y=10$
Amplitude 4; midline $y=8$

Show answer and explanation

Amplitude 4; midline

y=10

The amplitude is

(14-6)/2=4

. The midline is

(14+6)/2=10

.

Question 3

Which input is excluded from the domain of

y=\tan x

?

$x=\pi$
$x=0$
$x=\pi/2$
$x=2\pi$

Show answer and explanation

x=\pi/2

At

x=\pi/2

, cosine is zero, so tangent is undefined.

Key terms

Amplitude: The distance from a sinusoidal graph's midline to a maximum or minimum.
Midline: The horizontal line halfway between the maximum and minimum values of a sinusoidal graph.
Period: The horizontal length of one complete repeating cycle.
Vertical asymptote: A vertical line at an input where a function is undefined and its graph approaches the line.
Domain: The set of input values for which a function is defined.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Alberta Grade 12 Mathematics (Math 30-1), expectation TR4. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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TR4 · Graph sine, cosine, and tangent, and model with sinusoidal functions

Alberta Grade 12 Mathematics

Sine, cosine, tangent, and sinusoidal models

What you will learn

1. Prerequisites: angles and coordinates

2. Read the parent graphs

3. Transform and model sine and cosine

4. A reliable graphing process

Graph a transformed cosine function

Build a model for a repeating height

Read the parent tangent graph

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through Math 30-1

About this lesson and its review