Graph sine and cosine functions in radians

Ontario Grade 12 Mathematics

Trigonometric Functions

Use key points, amplitude, period, and shifts to sketch trigonometric graphs.

Sine and cosine graphs repeat in a regular wave pattern. To sketch one, you do not need to calculate every point. You can use a small set of key points, then adjust their positions and heights to match the function. This lesson focuses on graphs with angles measured in radians. It begins with the basic sine and cosine patterns, then uses one transformed function to show how the graph changes.

What you will learn

Recall the radian measures and key points needed to sketch sine and cosine graphs.
Identify the amplitude, period, horizontal shift, and vertical shift of a sine or cosine function.
Graph a sine or cosine function by plotting and connecting its key points.

1. Prerequisite bridge: radians and the basic patterns

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

360^\circ

, which is the same as

2\pi

radians. Half a turn is

\pi

, and a quarter turn is

\frac{\pi}{2}

. On a graph, radians are the input values along the horizontal axis.

The sine function is written

y=\sin x

, and the cosine function is written

y=\cos x

. Both have outputs from

-1

to

1

. Their graphs repeat after one full turn, so each has a period of

2\pi

. The period is the horizontal distance needed for one complete repetition.

A useful starting point is to plot five points across one period. For sine, the outputs rise from zero, reach one, return to zero, reach negative one, and return to zero. For cosine, the outputs begin at one, fall to zero, reach negative one, rise to zero, and return to one.

y=\sin x, y=\cos x

One complete cycle spans $2\pi$ radians.
The basic sine and cosine graphs both range from $-1$ to $1$ .
The five key inputs across one cycle are $0$ , $\frac{\pi}{2}$ , $\pi$ , $\frac{3\pi}{2}$ , and $2\pi$ .

2. Read the graph features

A transformed sine or cosine graph can be written as

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

or

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

. The same features apply to both forms. The coefficient

a

controls the vertical stretch or reflection. The amplitude is |a|, the distance from the centre line to a maximum or minimum. The centre line is the horizontal line halfway between those extremes.

The coefficient

k

changes the period. The period is

\frac{2\pi}{|k|}

. A larger value of |k| makes a cycle shorter; a smaller value makes it longer. The value

d

gives the horizontal shift: positive

d

moves the graph right, while negative

d

moves it left. The value

c

shifts the graph vertically, so the centre line is

y=c

.

The maximum and minimum values can be read from the centre line and amplitude. They are

c+|a|

and

c-|a|

. If

a

is negative, the graph is reflected vertically compared with the same function with positive

a

. Its key points still fit the same horizontal pattern, but the highs and lows switch.

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

Amplitude: |a|.
Period: $\frac{2\pi}{|k|}$ .
Centre line: $y=c$ ; maximum and minimum: $c+|a|$ and $c-|a|$ .
The value $d$ shifts the graph horizontally.

3. Build a graph from key points

For a sine graph, begin at the centre line when the inside angle is zero. Over one period, move through five equally spaced inputs. Their outputs follow the pattern centre, high, centre, low, centre when

a

is positive. For cosine, the pattern is high, centre, low, centre, high. A negative

a

reverses the high and low positions.

To use these patterns after a horizontal shift, start at

x=d

. Divide the period into four equal intervals. The five input values are

d

,

d+\frac{P}{4}

,

d+\frac{P}{2}

,

d+\frac{3P}{4}

, and

d+P

, where

P

is the period. At each input, use the sine or cosine pattern and place the output relative to the centre line.

Plot the five points and connect them with a smooth wave that follows the pattern. Extend the graph by repeating the cycle if the requested viewing window includes more than one period. A sketch should show the horizontal scale in radians and label important values such as

\pi

or fractions of

\pi

.

x=d,\ d+\frac{P}{4},\ d+\frac{P}{2},\ d+\frac{3P}{4},\ d+P

Find the period before choosing the five input values.
The key inputs are separated by one quarter of a period.
Use the centre line and amplitude to place the outputs.

4. Sketch carefully and interpret the result

A graph is easier to read when its axes are labelled clearly. Mark the horizontal axis in radians and show the centre line, maximum, and minimum on the vertical scale. Choose a scale that lets the important key inputs fit without crowding.

Before drawing the wave, check that the first point matches the function. A sine graph with no horizontal shift starts on its centre line. A cosine graph with no horizontal shift starts at a maximum when its coefficient is positive. If there is a shift, use the inside angle

k(x-d)

to locate where the cycle begins: it equals zero at

x=d

.

After one cycle is plotted, check the ending point. It should match the starting point because the graph has completed one period. Also check that the graph’s centre line and maximum and minimum agree with the values of

a

and

c

. These checks help catch errors in the scale, shift, and vertical placement.

k(x-d)=0 \Longrightarrow x=d

Label the horizontal axis in radians.
Use $x=d$ as the start of a cycle in the transformed form.
Check the first and last points, the centre line, and the extreme values.

Key points for one basic cycle

Input $x$	Sine output $\sin x$	Cosine output $\cos x$
$0$	$0$	$1$
$\frac{\pi}{2}$	$1$	$0$
$\pi$	$0$	$-1$
$\frac{3\pi}{2}$	$-1$	$0$
$2\pi$	$0$	$1$

Worked example

Sketch a shifted sine function

Sketch one complete cycle of

y=2\sin\left(2\left(x-\frac{\pi}{4}\right)\right)+1

.

Identify the features
Compare the function with the transformed sine form. The coefficient of sine gives an amplitude of $2$ . The inside coefficient is $2$ , so the period is $\frac{2\pi}{2}=\pi$ . The graph shifts right by $\frac{\pi}{4}$ , and its centre line is $y=1$ .
$A=2, P=\pi, d=\frac{\pi}{4}, c=1$
Find the key inputs
The first point occurs at $x=d=\frac{\pi}{4}$ . Divide the period by four to get an interval of $\frac{\pi}{4}$ . Add that interval repeatedly to find the five inputs for one cycle.
$\frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}, \pi, \frac{5\pi}{4}$
Place the key outputs
Because the function is a positive sine graph, its pattern is centre, high, centre, low, centre. The centre line is $1$ , and the amplitude is $2$ , so the high value is $3$ and the low value is $-1$ .
$(\frac{\pi}{4},1), (\frac{\pi}{2},3), (\frac{3\pi}{4},1), (\pi,-1), (\frac{5\pi}{4},1)$
Draw and check
Plot the five points and connect them with a smooth sine-shaped wave. The first and last points both lie on the centre line, and they are separated by one period, $\pi$ . The highest and lowest values are equally far from $y=1$ .
$y_{\max}=3, y_{\min}=-1$

Answer: One cycle runs from

x=\frac{\pi}{4}

to

x=\frac{5\pi}{4}

. It has centre line

y=1

, maximum

3

, and minimum

-1

, passing through the five listed key points.

Check: The horizontal distance from the first to last input is

\frac{5\pi}{4}-\frac{\pi}{4}=\pi

, which matches the period.

Common mistakes and how to avoid them

Using

2\pi

as the period even when the function contains an inside coefficient such as

k=3

.

Correction: Use

\frac{2\pi}{|k|}

to calculate the period, then divide that period into four equal intervals.

Moving a graph left when

d

is positive in

x-d

.

Correction: In the form

k(x-d)

, positive

d

shifts the graph right. Start the cycle at

x=d

.

Treating the vertical shift as the maximum value.

Correction: The vertical shift gives the centre line

y=c

. The maximum and minimum are

c+|a|

and

c-|a|

.

Spacing the five key points by the full period.

Correction: Divide the period by four. That gives the spacing between the five points across one cycle.

Lesson summary

The basic sine and cosine graphs repeat every $2\pi$ radians.
For transformed graphs, use amplitude |a|, period $\frac{2\pi}{|k|}$ , horizontal shift $d$ , and centre line $y=c$ .
Plot five equally spaced key points over one period, then connect them with a smooth wave.

Check your understanding

Question 1

What is the period of

y=\cos(3x)

?

$6\pi$
$2\pi$
$\frac{2\pi}{3}$
$\frac{\pi}{3}$

Show answer and explanation

\frac{2\pi}{3}

The inside coefficient is

3

. The period is

\frac{2\pi}{|3|}=\frac{2\pi}{3}

.

Question 2

For

y=4\sin(x-\pi)+2

, what is the centre line?

$y=4$
$y=2$
$y=-2$
$y=0$

Show answer and explanation

y=2

The vertical shift is

2

, so the centre line is

y=2

.

Question 3

For

y=\sin\left(x-\frac{\pi}{3}\right)

, where does the shifted cycle begin?

$x=0$
$x=-\frac{\pi}{3}$
$x=\frac{\pi}{3}$
$x=2\pi$

Show answer and explanation

x=\frac{\pi}{3}

Set the inside angle to zero. Then

x-\frac{\pi}{3}=0

, so the cycle begins at

x=\frac{\pi}{3}

.

Key terms

Amplitude: The distance from the centre line to a maximum or minimum of the graph.
Centre line: The horizontal line halfway between the graph’s maximum and minimum values.
Period: The horizontal length of one complete repeating cycle.
Key point: A point used to set the main shape and position of a trigonometric graph.

Continue through MHF4U

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 12 Mathematics (MHF4U), expectation B2.1. 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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