Sketch sine and cosine graphs and describe their properties

Ontario Grade 11 Mathematics

Trigonometric Functions

MCR3U • Unit D2 • Expectation D2.4

You already know how to find the sine and cosine of an angle in a right triangle. In Grade 10 you extended those ratios to any angle using the unit circle. In this lesson you will take the next step: treating sine and cosine as functions of an angle measured in degrees and drawing their full graphs. Once you can see the graphs, patterns that were invisible in a table — repeating waves, symmetry, predictable highs and lows — become obvious. Understanding these graphs is the foundation for everything else in the trigonometric functions strand of MCR3U.

What you will learn

Identify and describe the key properties of the sine and cosine functions, including amplitude, period, maximum, minimum, domain, and range.
Use a table of values to plot accurate graphs of y = sin x and y = cos x in degrees.
Read and interpret features of sine and cosine graphs, including x-intercepts, turning points, and symmetry.
Compare and contrast the shapes and positions of the sine and cosine graphs.

Prerequisite Bridge: Angles, the Unit Circle, and Function Notation

Recall from Grade 10 that any angle can be placed in standard position on a coordinate grid, with its vertex at the origin and its initial arm along the positive x-axis. A point on the terminal arm at distance 1 from the origin lies on the unit circle. If that point has coordinates

(x, y)

, then

\cos\theta = x

and

\sin\theta = y

, where

\theta

is the angle in degrees.

This means sine and cosine each produce a single output for every input angle. That is exactly the definition of a function. We write

f(\theta) = \sin\theta

and

g(\theta) = \cos\theta

. The input is the angle in degrees and the output is a ratio between

-1

and

1

— no unit circle point can go further than 1 unit from the origin.

Because the terminal arm returns to its starting position after a full CAD 360° rotation, the output values must repeat exactly every CAD 360°. This repetition is the most important feature we will study.

Sine and cosine outputs always satisfy $-1 \leq \sin\theta \leq 1$ and $-1 \leq \cos\theta \leq 1$ .
Both functions repeat their output pattern every CAD 360°.
Angles in this course are always measured in degrees.

Building the Graphs from a Table of Values

The clearest way to understand these graphs for the first time is to calculate outputs at key angles and then plot the points. The key angles are multiples of CAD 30° and CAD 45°, but for a clean first sketch the multiples of CAD 90° are enough: CAD 0°, 90°, 180°, 270°, 360°. These give you the highest point, the lowest point, and the x-intercepts of each wave.

For

f(\theta) = \sin\theta

: at

\theta = 0°

,

\sin 0° = 0

; at

\theta = 90°

,

\sin 90° = 1

; at

\theta = 180°

,

\sin 180° = 0

; at

\theta = 270°

,

\sin 270° = -1

; at

\theta = 360°

,

\sin 360° = 0

. Plot these five points and connect them with a smooth S-shaped curve. The curve rises from

0

, peaks at

1

, returns to

0

, dips to

-1

, and returns to

0

— one complete cycle.

For

g(\theta) = \cos\theta

: at

\theta = 0°

,

\cos 0° = 1

; at

\theta = 90°

,

\cos 90° = 0

; at

\theta = 180°

,

\cos 180° = -1

; at

\theta = 270°

,

\cos 270° = 0

; at

\theta = 360°

,

\cos 360° = 1

. Plot and connect. This curve starts at its maximum, descends through zero, reaches its minimum, and rises back to the maximum — one complete cycle.

Extend both graphs to the left (negative angles) and right (beyond CAD 360°) by repeating the same wave pattern. The graph continues forever in both directions.

Key angles at multiples of CAD 90° give the peaks, troughs, and zeros of each graph.
The sine graph starts at $(0°, 0)$ ; the cosine graph starts at $(0°, 1)$ .
Both complete one full cycle over any interval of length CAD 360°.
Both graphs extend infinitely in both directions.

Key Properties: Amplitude, Period, Domain, and Range

Amplitude is the distance from the midline of a wave to its highest (or lowest) point. The midline of both

y = \sin\theta

and

y = \cos\theta

is the x-axis, which has equation

y = 0

. The highest point on each graph is

1

and the lowest is

-1

. The distance from the midline to either extreme is

1

, so the amplitude of both functions is

1

.

Period is the length of one complete cycle, measured along the horizontal axis. Because both functions repeat every CAD 360°, the period of

y = \sin\theta

and

y = \cos\theta

is CAD 360°.

Domain is the set of all possible input values. Because any angle — positive, negative, or zero — can be placed in standard position, both functions accept any real-number angle in degrees. In interval notation the domain is all real numbers, which we write as \{\theta \mid \theta ∈

\mathbb{R}

\}.

Range is the set of all possible output values. Since sine and cosine outputs are always between

-1

and

1

inclusive, the range of both functions is

\{y \mid -1 \leq y \leq 1\}

.

Maximum value is

1

and minimum value is

-1

for both functions. The maximum of

\sin\theta

occurs at

\theta = 90° + 360°n

for any integer

n

, and the maximum of

\cos\theta

occurs at

\theta = 0° + 360°n

.

\text{Amplitude} = 1, \text{Period} = 360°

Amplitude of both $y = \sin\theta$ and $y = \cos\theta$ is $1$ .
Period of both functions is CAD 360°.
Domain: all real-number degree values.
Range: $-1 \leq y \leq 1$ for both functions.
Maximum value is $1$ ; minimum value is $-1$ .

Comparing Sine and Cosine: Similarities and Differences

Both graphs are smooth, continuous waves with the same amplitude, the same period, the same domain, and the same range. Their overall shape is identical — each looks like a rolling wave. In fact, if you slide the cosine graph CAD 90° to the right along the horizontal axis, it lines up perfectly with the sine graph. This is why the two functions are so closely related.

The key difference is where each graph starts at

\theta = 0°

. The sine graph passes through the origin

(0°, 0)

and is increasing at that point. The cosine graph starts at its maximum

(0°, 1)

and is decreasing. This starting position shifts all features: the x-intercepts, peaks, and troughs of cosine occur CAD 90° earlier than those of sine.

X-intercepts (where the graph crosses the horizontal axis) also differ. For

y = \sin\theta

, zeros occur at

0°, 180°, 360°, \ldots

and at

-180°, -360°, \ldots

— every multiple of CAD 180°. For

y = \cos\theta

, zeros occur at

90°, 270°, 450°, \ldots

and at

-90°, -270°, \ldots

— that is, at

90° + 180°n

for any integer

n

.

Both graphs have a line of symmetry about their own midline (

y = 0

): the upper half mirrors the lower half. The cosine graph is also symmetric about the vertical axis (

y

-axis), meaning

\cos(-\theta) = \cos\theta

. The sine graph has a rotational (point) symmetry about the origin, meaning

\sin(-\theta) = -\sin\theta

.

Sine and cosine have identical amplitude, period, domain, and range.
Cosine is shifted CAD 90° to the left compared to sine.
Sine zeros: every multiple of CAD 180°. Cosine zeros: $90° + 180°n$ .
$\cos(-\theta) = \cos\theta$ (even symmetry); $\sin(-\theta) = -\sin\theta$ (odd symmetry).

Reading the Graph: Intercepts, Turning Points, and Cycles

A turning point is a point where the graph changes from increasing to decreasing (a peak, called a local maximum) or from decreasing to increasing (a trough, called a local minimum). For

y = \sin\theta

, the peaks occur at

\theta = 90° + 360°n

with value

1

, and the troughs occur at

\theta = 270° + 360°n

with value

-1

. For

y = \cos\theta

, peaks occur at

\theta = 0° + 360°n

and troughs at

\theta = 180° + 360°n

.

A single cycle of each function is one complete repetition of the wave pattern. You can start a cycle at any point on the graph; the cycle ends exactly CAD 360° later at the same height and with the same direction of movement. Being able to identify where one cycle begins and ends lets you sketch any portion of the graph accurately.

When you are asked to sketch, always label: the scale on both axes, at least one full cycle (from a sensible start point to CAD 360° later), the coordinates of each peak, each trough, and each x-intercept within the sketched region. A rough freehand wave with no labels is not a complete sketch.

Peaks and troughs are called turning points; they are where the graph reverses direction.
One cycle spans exactly CAD 360° on the horizontal axis.
A complete sketch labels axes, scale, turning points, and x-intercepts.

Key Values of sin θ and cos θ at Multiples of 90°

θ (degrees)	sin θ	cos θ	Location on Wave
0°	0	1	Sine: zero (rising); Cosine: maximum
90°	1	0	Sine: maximum; Cosine: zero (falling)
180°	0	−1	Sine: zero (falling); Cosine: minimum
270°	−1	0	Sine: minimum; Cosine: zero (rising)
360°	0	1	Sine: zero (rising); Cosine: maximum

Comparing Properties of y = sin θ and y = cos θ

Property	y = sin θ	y = cos θ
Amplitude	1	1
Period	360°	360°
Domain	All real degrees	All real degrees
Range	−1 ≤ y ≤ 1	−1 ≤ y ≤ 1
Value at θ = 0°	0	1
Peaks occur at	90° + 360°n	0° + 360°n
Zeros occur at	0° + 180°n	90° + 180°n

Worked example

Example 1 — Completing a Table and Sketching y = sin θ

Complete the table of values for

y = \sin\theta

at

\theta = 0°, 30°, 60°, 90°, 120°, 150°, 180°, 210°, 240°, 270°, 300°, 330°, 360°

. Then describe the amplitude, period, domain, and range, and state the coordinates of every turning point and x-intercept in the interval

0° \leq \theta \leq 360°

.

Recall exact sine values at multiples of 30°
Use the special triangles you know from Grade 10. The CAD 30° $-$ 60° $-$ 90° triangle gives $\sin 30° = 0.5$ , $\sin 60° \approx 0.866$ . The unit-circle symmetry then fills in the rest. Round to two decimal places where needed.
$\sin 30° = 0.50, \sin 60° \approx 0.87$
Use symmetry to complete the second quadrant (90° to 180°)
Angles in the second quadrant share reference angles with first-quadrant angles, but sine stays positive there. So $\sin 120° = \sin 60° \approx 0.87$ and $\sin 150° = \sin 30° = 0.50$ . At the boundary, $\sin 180° = 0$ .
$\sin 120° \approx 0.87, \sin 150° = 0.50, \sin 180° = 0$
Apply the sign rule for the third and fourth quadrants
Sine is negative in the third quadrant (CAD 180° to CAD 270°) and negative in the fourth (CAD 270° to CAD 360°). The magnitudes mirror the first and second quadrants. So $\sin 210° = -0.50$ , $\sin 240° \approx -0.87$ , $\sin 270° = -1$ , $\sin 300° \approx -0.87$ , $\sin 330° = -0.50$ , $\sin 360° = 0$ .
$\sin 270° = -1, \sin 330° = -0.50$
Plot the points and draw a smooth wave
Set up axes with $\theta$ on the horizontal axis (label CAD 0° to CAD 360° in steps of CAD 30°) and $y$ on the vertical axis (label from $-1$ to $1$ ). Plot all 13 points from the table. Connect them with a single smooth curve — not straight line segments. The result is one complete sine wave.
State the properties
The highest point on the graph is $y = 1$ and the lowest is $y = -1$ . The distance from the midline ( $y = 0$ ) to either extreme is $1$ , so the amplitude is $1$ . The graph completes one full cycle from CAD 0° to CAD 360°, so the period is CAD 360°. The domain is all real-number angles in degrees; the range is $-1 \leq y \leq 1$ .
$\text{Amplitude} = 1, \text{Period} = 360°$
Identify turning points and x-intercepts
The peak (maximum turning point) occurs at $(90°, 1)$ . The trough (minimum turning point) occurs at $(270°, -1)$ . The graph crosses the x-axis — meaning $\sin\theta = 0$ — at $(0°, 0)$ , $(180°, 0)$ , and $(360°, 0)$ .
$\text{Peak: } (90°, 1); \text{Trough: } (270°, -1)$

Answer: Amplitude

= 1

; Period

= 360°

; Domain: all real numbers (degrees); Range:

-1 \leq y \leq 1

. In

0° \leq \theta \leq 360°

: peak at

(90°, 1)

, trough at

(270°, -1)

, x-intercepts at

\theta = 0°, 180°, 360°

.

Check: Verify three values directly:

\sin 90° = 1

✓ (unit circle top);

\sin 270° = -1

✓ (unit circle bottom);

\sin 180° = 0

✓ (terminal arm on negative x-axis). All turning points and intercepts are confirmed.

Worked example

Example 2 — Sketching y = cos θ and Comparing with y = sin θ

Sketch

y = \cos\theta

for

-90° \leq \theta \leq 450°

. On the same grid, lightly sketch

y = \sin\theta

. State: (a) all x-intercepts of the cosine graph in the given interval, (b) all turning points of the cosine graph in the given interval, and (c) one similarity and one difference between the two graphs.

Set up the table for cosine at key angles
Use multiples of CAD 90° $within$ -90° to CAD 450°: that is $-90°, 0°, 90°, 180°, 270°, 360°, 450°$ . Evaluate $\cos\theta$ at each. Recall that cosine equals the x-coordinate of the unit circle point.
$\cos(-90°) = 0, \cos 0° = 1, \cos 90° = 0, \cos 180° = -1, \cos 270° = 0, \cos 360° = 1, \cos 450° = 0$
Plot and connect the cosine points
Label the horizontal axis from $-90°$ to CAD 450° in steps of CAD 90°. Label the vertical axis from $-1$ to $1$ . Plot the seven points from Step 1. Connect with a smooth continuous wave. The graph starts at $(- 90°, 0)$ , rises to a peak at $(0°, 1)$ , drops to a trough at $(180°, -1)$ , rises back to a peak at $(360°, 1)$ , and ends at $(450°, 0)$ .
Lightly add the sine graph for comparison
On the same axes, sketch $y = \sin\theta$ . Its key points in the interval are: $(-90°, -1)$ , $(0°, 0)$ , $(90°, 1)$ , $(180°, 0)$ , $(270°, -1)$ , $(360°, 0)$ , $(450°, 1)$ . Notice the sine graph is $0$ where cosine is at its maximum, and vice versa.
List the x-intercepts of the cosine graph
X-intercepts occur where $\cos\theta = 0$ . From the plotted points and the known pattern, these occur at $-90°, 90°, 270°,$ and CAD 450° in the given interval. Check: all are of the form $90° + 180°n$ for integer $n$ .
$\theta = -90°, 90°, 270°, 450°$
List the turning points of the cosine graph
The peaks (local maxima) occur where $\cos\theta = 1$ : at $(0°, 1)$ and $(360°, 1)$ . The troughs (local minima) occur where $\cos\theta = -1$ : at $(180°, -1)$ . These are all the turning points visible in the interval $-90°$ to CAD 450°.
$\text{Peaks: } (0°, 1), (360°, 1); \text{Trough: } (180°, -1)$
Compare the two graphs
Similarity: Both $y = \sin\theta$ and $y = \cos\theta$ have amplitude $1$ , period CAD 360°, the same domain, and the same range $-1 \leq y \leq 1$ . Their wave shapes are identical. Difference: The cosine graph reaches its maximum at $\theta = 0°$ , while the sine graph is at $0$ and increasing at $\theta = 0°$ . Equivalently, the cosine graph is shifted CAD 90° to the left of the sine graph.

Answer: (a) X-intercepts of

y = \cos\theta

in

[-90°, 450°]

:

\theta = -90°, 90°, 270°, 450°

. (b) Turning points: peaks at

(0°, 1)

and

(360°, 1)

; trough at

(180°, -1)

. (c) Similarity: both have amplitude

1

and period CAD 360°. Difference: cosine starts at its maximum

(0°, 1)

while sine starts at the origin

(0°, 0)

.

Check: Spot-check with the unit circle:

\cos 0° = 1

✓ (point

(1,0)

);

\cos 180° = -1

✓ (point

(-1,0)

);

\cos 90° = 0

✓ (point

(0,1)

). The shift relationship is confirmed because

\sin(\theta + 90°) = \cos\theta

, consistent with what the graphs show.

Common mistakes and how to avoid them

Connecting plotted points with straight line segments instead of a smooth curve, producing a zigzag rather than a wave.

Correction: Sine and cosine are smooth functions. Always connect points with a gently curving wave. The curve should have no sharp corners.

Stating that the period is 180° because the graph seems to repeat a half-wave pattern.

Correction: One period is one complete cycle — up AND down. A half-wave is not a full repetition. The period of both y = sin θ and y = cos θ is 360°.

Forgetting to include negative angles when asked for a graph that extends to the left of 0°, and leaving that part of the axes blank.

Correction: The graph continues to the left of 0° by repeating the same wave pattern. At θ = −90°, sin(−90°) = −1 and cos(−90°) = 0, and so on.

Mixing up which function has its maximum at θ = 0° and which has a zero at θ = 0°.

Correction: Cosine equals 1 at θ = 0° (it starts at its peak). Sine equals 0 at θ = 0° (it starts at the midline). Remember: C for Cosine, C for Crest at zero degrees.

Writing the range as 0 ≤ y ≤ 1 because only positive outputs are visible in the first quadrant.

Correction: Sine and cosine are negative in the third and fourth quadrants. The full range is −1 ≤ y ≤ 1 for both functions.

Lesson summary

The sine and cosine functions take an angle in degrees as input and produce a value between −1 and 1 as output.
Both graphs are smooth, continuous waves with amplitude 1 and period 360°.
The domain of each function is all real-number degree values; the range is −1 ≤ y ≤ 1.
The sine graph passes through the origin (0°, 0) and rises; the cosine graph starts at its peak (0°, 1) and falls. This 90° shift in starting position is the key difference between the two graphs.
To sketch accurately: build a table at multiples of 90° (or 30°), plot the points, join with a smooth wave, and label axes, scale, turning points, and x-intercepts.
Turning points for sine occur at 90° + 360°n (peaks) and 270° + 360°n (troughs); for cosine they occur at 0° + 360°n (peaks) and 180° + 360°n (troughs), for any integer n.

Check your understanding

Question 1

What is the range of the function

y = \cos\theta

?

All real numbers
$0 \leq y \leq 1$
$-1 \leq y \leq 1$
$-90° \leq y \leq 90°$

Show answer and explanation

-1 \leq y \leq 1

The cosine function produces outputs only between −1 and 1 inclusive, because it represents the x-coordinate of a point on the unit circle. The range is −1 ≤ y ≤ 1.

Question 2

At which angle does

y = \sin\theta

reach its minimum value of

-1

?

$\theta = 0°$
$\theta = 90°$
$\theta = 180°$
$\theta = 270°$

Show answer and explanation

\theta = 270°

The minimum of the sine function is −1, which occurs at θ = 270°. At this angle the unit circle point is at the very bottom, giving y-coordinate −1. Verify: sin 270° = −1 ✓.

Question 3

How many full cycles does

y = \sin\theta

complete between

\theta = -360°

and

\theta = 360°

?

1
2
3
4

Show answer and explanation

2

The period is 360°. The interval from −360° to 360° has a total length of 720°. Dividing: 720° ÷ 360° = 2. So exactly 2 full cycles are completed.

Question 4

Which statement correctly describes a difference between the graphs of

y = \sin\theta

and

y = \cos\theta

?

They have different amplitudes.
The cosine graph has a greater maximum value than the sine graph.
The cosine graph reaches its maximum at $\theta = 0°$ , while the sine graph equals 0 at $\theta = 0°$ .
They have different periods.

Show answer and explanation

The cosine graph reaches its maximum at

\theta = 0°

, while the sine graph equals 0 at

\theta = 0°

.

Both functions have the same amplitude (1), the same maximum value (1), and the same period (360°). The real difference is the starting position: cos 0° = 1 (its maximum) while sin 0° = 0 (crossing the midline). The cosine graph is shifted 90° to the left compared to sine.

Key terms

Amplitude: The distance from the midline of a wave to its highest or lowest point. For y = sin θ and y = cos θ, the amplitude is 1.
Period: The horizontal length of one complete cycle of a repeating graph. For y = sin θ and y = cos θ, the period is 360°.
Cycle: One complete repetition of a wave pattern, from any starting point back to the same height and direction of movement.
Turning point: A point on a curve where the graph changes from increasing to decreasing (a peak) or from decreasing to increasing (a trough).
Amplitude: The distance from the midline to the maximum (or minimum) of the graph.
Midline: The horizontal axis about which a wave oscillates. For y = sin θ and y = cos θ, the midline is y = 0.
Domain: The complete set of all allowed input values for a function. For sine and cosine, the domain is all real-number angles in degrees.
Range: The complete set of all possible output values of a function. For sine and cosine, the range is −1 ≤ y ≤ 1.

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