Sketch the sine graph and describe its properties

Ontario Grade 11 Mathematics

Trigonometric Functions

The shape and key properties of $y=\sin x$

A sine graph shows how the sine value changes as an angle changes. Its pattern repeats, rising and falling in a smooth wave. In this lesson, we focus on the basic sine function,

y=\sin x

, with

x

measured in degrees. You will use a small set of familiar angles to sketch the graph and describe its main properties.

What you will learn

Use familiar angle values to find points on the sine graph.
Sketch one cycle of $y=\sin x$ by plotting key points and joining them with a smooth curve.
Describe the domain, range, intercepts, maximum, minimum, and period of the sine function.

1. Prerequisite bridge: angles and coordinates

A point on a graph is written as an ordered pair,

(x,y)

. The first number tells you where to move along the horizontal axis. The second tells you where to move along the vertical axis.

The sine of an angle is a number between

-1

and

1

. For the basic sine graph, the angle is the input

x

, and its sine is the output

y

. For example,

\sin 0^\circ=0

, so the graph includes the point

(0,0)

.

You may already know the sine values of special angles from right-triangle trigonometry. Here, those values become points on a graph. A degree symbol shows that an angle is measured in degrees.

y=\sin x

The graph of $y=\sin x$ uses the angle as its input and the sine value as its output.
Sine values stay between $-1$ and $1$ .

2. The shape of one sine cycle

A cycle is one complete repeat of a graph’s pattern. From

0^\circ

to

360^\circ

, the sine graph starts at zero, rises to

1

, returns to zero, falls to

-1

, and returns to zero.

The five key angles divide the cycle into four equal parts. Plot their sine values as coordinates. Then join the points with a smooth curve. The curve does not have sharp corners at the high and low points.

The highest point is called a maximum. The lowest point is called a minimum. The vertical distance from the middle level,

y=0

, to either extreme is

1

for this basic graph. This distance is its amplitude.

The period is the horizontal length of one complete repeat. For

y=\sin x

, the period is

360^\circ

. After that, the same pattern begins again.

\sin 0^\circ=0,\quad \sin 90^\circ=1,\quad \sin 180^\circ=0,\quad \sin 270^\circ=-1,\quad \sin 360^\circ=0

One cycle runs from $0^\circ$ to $360^\circ$ .
The maximum value is $1$ and the minimum value is $-1$ .
The amplitude is $1$ and the period is $360^\circ$ .

3. Representations and properties

The table lists the key points for one cycle. Each row gives an angle and its sine value. Read each row as a coordinate, with the angle first and the sine value second.

The graph crosses the horizontal axis whenever its value is zero. These crossings are called

x

-intercepts. In one cycle from

0^\circ

to

360^\circ

, they occur at

0^\circ

,

180^\circ

, and

360^\circ

.

The domain is the set of allowed input values. The sine function accepts any real angle, so its domain is all real numbers. The range is the set of possible output values. Its range is from

-1

to

1

, including both ends.

The graph keeps repeating to the left and right. In addition, its values reflect across the origin: for example, the value at

-90^\circ

is the opposite of the value at

90^\circ

. These patterns can help you extend a sketch beyond one cycle.

\sin(x+360^\circ)=\sin x

Domain: all real values of $x$ .
Range: $-1\leq y\leq 1$ .
The graph repeats every $360^\circ$ .
In one cycle, the $x$ -intercepts are $(0^\circ,0)$ , $(180^\circ,0)$ , and $(360^\circ,0)$ .

4. Sketching carefully

Start by marking the horizontal axis in degrees and the vertical axis from

-1

to

1

. Plot the five key points from the table. Check that the graph rises from the first point to the maximum, falls through the next intercept to the minimum, and rises back to the final point.

A sketch should show the correct shape and important features, even if it is not drawn to exact scale. Label the axes and include the key values. If you extend the graph, repeat the same wave every

360^\circ

.

Do not connect the points with straight line segments. The sine graph changes smoothly. Straight segments would make corners and would not show the correct graph.

Mark key angles and values before drawing the curve.
Use a smooth wave through the plotted points.
Check the maximum, minimum, intercepts, and repeat length.

Key points for one cycle

Angle, $x$	Sine value, $y$	Point, $(x,y)$
$0^\circ$	$0$	$(0^\circ,0)$
$90^\circ$	$1$	$(90^\circ,1)$
$180^\circ$	$0$	$(180^\circ,0)$
$270^\circ$	$-1$	$(270^\circ,-1)$
$360^\circ$	$0$	$(360^\circ,0)$

Worked example

Sketch one cycle and describe it

Sketch

y=\sin x

for

0^\circ\leq x\leq 360^\circ

, then state its maximum, minimum, intercepts, range, and period.

Choose key angles
Use the beginning, quarter-points, and end of the cycle. The sine values at these familiar angles provide enough points to show the main shape.
$x=0^\circ,\ 90^\circ,\ 180^\circ,\ 270^\circ,\ 360^\circ$
Plot the points
Pair each angle with its sine value. Plot the resulting coordinates on axes labelled in degrees horizontally and from $-1$ to $1$ vertically.
$(0^\circ,0),\ (90^\circ,1),\ (180^\circ,0),\ (270^\circ,-1),\ (360^\circ,0)$
Draw and describe the curve
Join the points with a smooth wave. It rises to $1$ , falls through zero to $-1$ , and rises back to zero. The largest and smallest outputs give the range, while the horizontal distance from the start to the end gives the period.
$-1\leq y\leq 1,\quad P=360^\circ$

Answer: The maximum is

1

at

90^\circ

. The minimum is

-1

at

270^\circ

. The intercepts in the stated interval are

(0^\circ,0)

,

(180^\circ,0)

, and

(360^\circ,0)

. The range is

[-1,1]

, and the period is

360^\circ

.

Check: The sketch begins and ends at zero, and it contains one high point and one low point. Those features match one complete sine cycle.

Common mistakes and how to avoid them

Drawing straight segments between the key points.

Correction: Draw a smooth curve through the points. The sine graph has no sharp corners.

Saying the range is from

0

to

1

because sine values are positive for part of the cycle.

Correction: Include the negative part of the cycle. The full range is from

-1

to

1

.

Calling the amplitude the distance from the maximum to the minimum.

Correction: For the basic sine graph, the amplitude is the distance from the middle level,

y=0

, to the maximum or minimum. It is

1

.

Confusing the period with the number of intercepts.

Correction: The period is a horizontal distance. One full repeat takes

360^\circ

.

Lesson summary

Use the sine values at $0^\circ$ , $90^\circ$ , $180^\circ$ , $270^\circ$ , and $360^\circ$ to plot one cycle.
Join the key points with a smooth wave.
For $y=\sin x$ , the amplitude is $1$ , the range is $[-1,1]$ , and the period is $360^\circ$ .
The graph crosses zero at $0^\circ$ , $180^\circ$ , and $360^\circ$ in one cycle.

Check your understanding

Question 1

For

y=\sin x

, what is the value of

y

at

x=270^\circ

?

$1$
$0$
$-1$
$360$

Show answer and explanation

-1

The key point at

270^\circ

is

(270^\circ,-1)

, the minimum of the cycle.

Question 2

Which statement gives the period of

y=\sin x

?

The graph repeats every $180^\circ$ .
The graph repeats every $360^\circ$ .
The graph repeats every $1^\circ$ .
The graph never repeats.

Show answer and explanation

The graph repeats every

360^\circ

.

One complete cycle runs from

0^\circ

to

360^\circ

, so the period is

360^\circ

.

Question 3

Which interval describes the range of

y=\sin x

?

$0\leq y\leq 1$
$-1\leq y\leq 0$
$-1\leq y\leq 1$
$y\geq 1$

Show answer and explanation

-1\leq y\leq 1

The graph reaches a maximum of

1

and a minimum of

-1

, so all output values lie between them, including both.

Key terms

Cycle: One complete repeat of a graph’s pattern.
Amplitude: For the basic sine graph, the distance from the middle level to its maximum or minimum.
Period: The horizontal length of one complete repeat.
Domain: The set of allowed input values.
Range: The set of possible output values.
Intercept: A point where a graph meets an axis. An $x$ -intercept has a $y$ -value of zero.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Mathematics (MCF3M), expectation C2.4. 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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