Year 11 Extended · Unit 2 Trigonometric functions and any triangle · Lesson 2 of 6

Two tuning forks make graphs of the same shape. What differs: the height, the spacing, or both?

Plot sine against the angle and a wave appears. Two numbers control it: how high it rises, and how long one wave takes. Both are transformations you met in Year 10.

Geometry and TrigonometryGDC optional

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From Lesson 1: which quadrant has sin positive and cos negative?
    AnswerThe second quadrant, 90° to 180°.
  2. From Year 10: the graph of y = f(x) is stretched vertically by a factor of 3. What is the new equation?
    Answery = 3f(x). Every y-value is multiplied by 3.
  3. From Lesson 1: write sin 270° and cos 270°.
    Answer−1 and 0. The tip is at (0, −1).
Words used in this lesson
amplitude
the distance from the middle line of a wave to its highest point. For example: the amplitude of y = 2 sin x is 2.
period
the length of one complete wave on the horizontal axis. For example: 360° for y = sin x.
periodic
repeating exactly after a fixed interval. For example: a wave that repeats.
midline
the horizontal line halfway between the highest and lowest values. For example: the x-axis for y = sin x.
asymptote
a line that a graph gets closer to but never reaches. For example: the x-axis for y = 1 ÷ x, far from the origin.

Two tuning forks

In a music school in Seoul, a microphone records the air pressure beside a tuning fork. Plotted against time, it makes a wave. Strike the fork harder and the wave rises higher; use a fork of a higher pitch and the waves come closer together. Our graphs do the same, with the angle in degrees on the horizontal axis.

Predict first: if you take the graph of y = sin x and change it to y = 3 sin x, what happens to the highest point, and what happens to the spacing of the peaks? Then if you change it to y = sin 4x, which of the two changes? Use the first figure to test your predictions, and the second to compare sine with cosine.

1
1
1amplitude
360period (degrees)

y = sin x: amplitude 1, period 360°. Highest value 1, lowest −1, 1 wave between 0° and 360°.

Show me the steps
  1. a = 1, b = 1: this is y = sin x. It rises to 1 at 90°, falls to −1 at 270°, and has one wave in 360°, so the period is 360°.
  2. Set a = 3, leaving b = 1: the graph is the same wave stretched vertically by a factor of 3. It reaches 3 and −3 but still repeats every 360°. The amplitude changed; the period did not.
  3. Set a = 2 and b = 4: there are 4 waves in 360°, so one wave takes 360° ÷ 4 = 90°. The graph is squeezed horizontally by a factor of 4 and stretched vertically by 2, and reaches 2 and −2.

The graph of y = a sin bx (with a and b positive) is a wave through the origin.

  • Amplitude: the distance from the middle line to a peak is a. The highest value is a and the lowest is −a. In Year 10 terms, a is a vertical stretch by factor a.
  • Period: one complete wave takes 360° ÷ b, so there are b waves in 360°. In Year 10 terms, replacing x by bx stretches horizontally by factor 1 ÷ b.
  • Cosine: y = cos x has the same shape, but starts at its highest point: cos x = sin (x + 90°), the sine moved 90° to the left.
  • A negative a turns the graph upside down (a reflection in the x-axis); the amplitude is still the size of a.
75
0.97sin x
0.26cos x

x = 75°: sin x = 0.97 and cos x = 0.26. The cosine at x equals the sine at x + 90°: sin 165° = 0.26.

Show me the steps
  1. x = 75°: sin x = 0.97 and cos x = 0.26. The sine at 165° (75° + 90°) is also 0.26, the cosine at 75°.
  2. x = 90°: sin x = 1 is the top of the sine wave and cos x = 0 is the point where the cosine crosses the axis. The cosine has its top at 0°, 90° earlier than the sine.
  3. x = 270°: sin x = −1 is the bottom of the sine wave, and cos x = 0 again. Both curves have period 360° and the same amplitude, 1; they differ only by a shift of 90°.

b counts the waves; it is not the length of a wave. In y = sin 4x there are 4 waves in 360°, so each wave is 90° long, not 4° long. A bigger b makes the waves narrower, and the period is 360° ÷ b, never 360° × b.

Worked examples

For y = 4 sin 3x, state the amplitude, the period and the highest and lowest values.

  1. The amplitude is the number in front: 4. So the highest value is 4 and the lowest is −4.
  2. The period is 360° ÷ 3 = 120°, so there are 3 waves between 0° and 360°.

Write an equation of a sine graph with amplitude 5 and period 90°.

  1. The amplitude gives a = 5.
  2. The period 90° = 360° ÷ b, so b = 360 ÷ 90 = 4. The equation is y = 5 sin 4x.

Your go. The first step is given. A cosine graph has highest value 2.5 and completes 3 waves between 0° and 360°. Write an equation for it.

  1. It is a cosine graph with amplitude 2.5, so y = 2.5 cos bx.
  2. Three waves in 360° gives the value of b directly.
Check the answer

b is the number of waves in 360°, so b = 3 and the equation is y = 2.5 cos 3x. Its period is 360° ÷ 3 = 120°. A common slip is to write b = 120, the period.

With a GDC (optional)

Set the angle mode to degrees before you graph anything. Enter y = a sin(bx) in a function slot, choose a window with x from 0 to 360 and y a little beyond the amplitude, and graph it. The maximum tool gives the highest point; the distance between two neighbouring peaks, read with trace, is the period.

Use the GDC to check the graph you sketched, not to replace the sketch: you should know before pressing the key how high the graph goes and where it first returns to its starting value.

Check before you go on

1. The graph of y = 3 sin x is changed to y = 3 sin 12x. What changes?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. State the period, in degrees, of the graph y = 6 sin 15x.

3. The graph of y = a sin x, with a positive, has a lowest value of −7. Find a.

Level 2 · Building

4. A sound wave is modelled by y = 2.5 sin bx, with x in degrees, and one complete wave takes 45°. Find b.

5. For y = 3 cos 4x, find the value of y when x = 45°.

Level 3 · Stretch

6. The graph of y = 5 sin bx first reaches its highest point when x = 30°. Find b.

7. The graph of y = 5 sin bx has a period of 40°. Find y when x = 5°, to 2 decimal places.

Something to think about

The graph of y = tan x is not a wave of fixed height. Using tan x = sin x ÷ cos x and what you know about where cos x = 0, what do you expect its graph to do near 90° and 270°, and what is its period?

One way to think about it

Near 90° and 270° the cosine is close to 0, so the tangent grows without limit: the graph has vertical asymptotes there. It repeats every 180°, because tan (x + 180°) = tan x, so its period is 180° and it has no amplitude.

Exit ticket

8. Which equation has a highest value of 1.5, a lowest value of −1.5 and 10 complete waves between 0° and 360°?