Grade by Grade
D. Trigonometric Functions · Sinusoidal Functions

The Graphs of y = sin x and y = cos x

D2.4About 70 minutes
Learning goals
  • I can sketch y = sin x and y = cos x quickly using five key points.
  • I can state the domain, range, amplitude, period, maximum and minimum points, zeros, y-intercept and axis of each graph.
  • I can state the intervals where each graph is increasing or decreasing.
  • I can describe how the graphs of sine and cosine are related.

Words to know

key points
The five points that divide one cycle into four equal parts: for y = sin x they are (0°, 0), (90°, 1), (180°, 0), (270°, −1) and (360°, 0).
zero
An x-value where the graph crosses or touches the x-axis (y = 0).
interval of increase
An interval of x-values where the graph rises as x increases.
interval of decrease
An interval of x-values where the graph falls as x increases.
parent function
The simplest function of a family, here y = sin x or y = cos x, which other members are transformed from.

Meet the parent functions

In the last lesson you built the sine wave point by point from the unit circle. From now on we will use x for the angle (in degrees) and write y = sin x and y = cos x. These are the parent functions of every sinusoidal function you will meet in this unit, just as y = x² was the parent of every parabola in Grade 10. Knowing their shapes and key properties by heart will make everything else much easier.

Five key points

One cycle of each graph splits into four equal quarters of 90°. At the ends of the quarters are the key points:

  • y = sin x: (0°, 0), (90°, 1), (180°, 0), (270°, −1), (360°, 0). Pattern: zero, max, zero, min, zero.
  • y = cos x: (0°, 1), (90°, 0), (180°, −1), (270°, 0), (360°, 1). Pattern: max, zero, min, zero, max.

To sketch a graph, plot the five points, join them with a smooth rounded curve (rounded at the top and bottom, not pointed), then repeat the cycle to the left and right.

−360°−270°−180°−90°0°90°180°270°360°−1012(90°, 1)(180°, 0)(270°, −1)(0, 0)decreasing from 90° to 270°y = sin x
y = sin x for −360° ≤ x ≤ 360°. One cycle from 0° to 360° shows the five key points. The shaded strip is where the graph is decreasing (90° < x < 270°).
−360°−270°−180°−90°0°90°180°270°360°−1012(0, 1)(90°, 0)(180°, −1)(270°, 0)(360°, 1)increasing from 180° to 360°y = cos x
y = cos x for −360° ≤ x ≤ 360°. It starts at its maximum (0, 1). The shaded strip is where it is increasing (180° < x < 360°).
Propertyy = sin xy = cos x
Domain{x ∈ ℝ}{x ∈ ℝ}
Range{y ∈ ℝ | −1 ≤ y ≤ 1}{y ∈ ℝ | −1 ≤ y ≤ 1}
Period360°360°
Amplitude11
Equation of the axisy = 0y = 0
Maximum value 1 atx = 90°, 450°, … (90° + 360°n)x = 0°, 360°, … (360°n)
Minimum value −1 atx = 270°, 630°, … (270° + 360°n)x = 180°, 540°, … (180° + 360°n)
Zerosx = 0°, 180°, 360°, … (180°n)x = 90°, 270°, … (90° + 180°n)
y-intercept01
Increasing (0° to 360°)0° < x < 90° and 270° < x < 360°180° < x < 360°
Decreasing (0° to 360°)90° < x < 270°0° < x < 180°
Key properties of the parent functions (n is any integer).
Worked example

Example 1: Reading properties from the graph

For y = sin x on −360° ≤ x ≤ 360°, state (a) the zeros, (b) the maximum and minimum points, (c) the intervals of increase and decrease.

  1. (a) The graph crosses the x-axis every 180°: x = −360°, −180°, 0°, 180°, 360°.
  2. (b) Maximum points: (−270°, 1) and (90°, 1). Minimum points: (−90°, −1) and (270°, −1).
  3. (c) The graph rises from each minimum to the next maximum. Increasing: −360° < x < −270°, −90° < x < 90° and 270° < x < 360°.
  4. It falls from each maximum to the next minimum. Decreasing: −270° < x < −90° and 90° < x < 270°.

Zeros every 180°; max at −270° and 90°; min at −90° and 270°; intervals as listed.

Worked example

Example 2: Solving with the graph

Use the graphs to find all x between 0° and 360° for which (a) sin x = 0.5 (b) cos x = −1 (c) cos x = 0.5.

  1. (a) Draw the line y = 0.5 across the sine graph. It meets the first hump twice, at x = 30° and x = 150° (sin 30° = 1/2 and the hump is symmetric about 90°).
  2. (b) cos x = −1 at the minimum of the cosine graph: x = 180° only.
  3. (c) y = 0.5 meets the cosine graph at x = 60° (cos 60° = 1/2) and, by symmetry about 180°, at 360° − 60° = 300°.
  4. These match the two-angle answers you found with related acute angles in Trigonometric Ratios and Laws.

(a) 30°, 150° (b) 180° (c) 60°, 300°

Sine and cosine are the same wave

Put the two graphs on the same axes. The cosine graph is exactly the sine graph moved 90° to the left. For example, the sine maximum at (90°, 1) moves to the cosine maximum at (0°, 1). In symbols:

cos x = sin(x + 90°) and sin x = cos(x − 90°)

So there is really just one wave shape. That is why both are called sinusoidal functions.

−90°0°90°180°270°360°450°−10190° left90° leftsolid: y = sin xdashed: y = cos x
Sliding the sine graph (solid) 90° to the left lands it exactly on the cosine graph (dashed).

Symmetry you can see

The cosine graph is a mirror image of itself in the y-axis, so cos(−x) = cos x. The sine graph has half-turn symmetry about the origin, so sin(−x) = −sin x. You can check both on the unit circle: x and −x are reflections of each other in the x-axis.

Just for fun

Why does the cosine graph always look so confident at x = 0?

Because it starts at the top, with a 90° head start on sine.

Hands-on activity

Slide, match and flip

You need: graph paper, a pencil, a ruler and a sheet of tracing paper (baking parchment works).

  1. On graph paper, let 1 square = 30° across and 4 squares = 1 unit up. Draw axes from −360° to 360°.
  2. Sketch y = sin x using the five key points for each cycle. Sketch y = cos x on the same axes in a different style (dashed).
  3. Trace the sine graph and the axes onto the tracing paper.
  4. Slide the tracing paper left along the x-axis until the traced sine graph lies on top of the cosine graph. How far did you slide it?
  5. Now try sliding it right instead. Find a slide to the right that also works.
  6. Flip the tracing paper over (turn it face down, top edge stays on top) and line up the axes. What graph do you see now?

What to look for: sliding 90° left (or 270° right) turns the sine graph into the cosine graph. Flipping it over left-to-right gives the graph of y = sin(−x), which is the same as y = −sin x (the sine graph upside down).

Your turn

1

State the domain, range, amplitude and period of y = cos x.

2

List the zeros of y = sin x for −360° ≤ x ≤ 720°.

3

Where does y = cos x reach its maximum value for −360° ≤ x ≤ 720°? What is that value?

4

State the intervals where y = sin x is increasing for 0° ≤ x ≤ 360°.

5

Use the graph to solve cos x = 0 for 0° ≤ x ≤ 360°.

6

For which x between 0° and 360° are both sin x and cos x positive?

7

Describe a translation that maps y = sin x onto y = cos x.

More practice: warm-up

1

State the y-intercepts of y = sin x and y = cos x.

2

What is the minimum value of y = sin x, and the first positive x where it happens?

3

Write the range of y = sin x.

4

What is the period of y = cos x?

5

What is the amplitude of y = sin x?

More practice: on your own

1

Solve sin x = −1 for −360° ≤ x ≤ 360°.

2

Sketch y = sin x and y = cos x on the same axes for 0° ≤ x ≤ 360°. Where do they cross?

3

State the intervals where y = cos x is decreasing for 0° ≤ x ≤ 720°.

4

How many solutions does sin x = 0.8 have for 0° ≤ x ≤ 720°? Explain.

5

Use the graph and exact values to solve cos x = −0.5 for 0° ≤ x ≤ 360°.

Challenge

1

Use the unit circle to explain why cos(−x) = cos x. What does this tell you about the graph of y = cos x?

2

Find all intersection points of y = sin x and y = cos x for −360° ≤ x ≤ 360°.

3

Consider y = |sin x| (the absolute value makes every negative y positive). Is it periodic? What is its period and range? Is it sinusoidal?

Check your understanding

Quiz

1.

What is the range of y = sin x?

2.

What are the zeros of y = cos x for 0° ≤ x ≤ 360°?

3.

On which interval is y = sin x decreasing?

4.

What is the y-intercept of y = cos x?

5.

Which transformation maps the graph of y = sin x onto the graph of y = cos x?

6.

How many maximum points does y = sin x have for 0° ≤ x ≤ 720°?

For parents and teachers

Big idea: D2.4 asks students to sketch y = sin x and y = cos x (degrees) and describe their key properties: cycle, domain, range, intercepts, amplitude, period, maximum and minimum values, and intervals of increase and decrease.

How to help: Ask your teen to sketch both graphs from memory in under a minute using the five key points ("zero, max, zero, min, zero" for sine; "max, zero, min, zero, max" for cosine). Then ask one quick property question: "Where is cosine increasing?"

Common difficulties:

  • Drawing pointed peaks like a zigzag. The graph is rounded at every maximum and minimum.
  • Mixing up the two starting points: sine starts at 0, cosine at 1.
  • Writing intervals with the wrong endpoints. Have your teen trace the graph with a finger from left to right, saying "up" or "down".
  • Forgetting that the graphs continue forever in both directions (domain all real numbers).