Year 11 Extended · Unit 2 Trigonometric functions and any triangle · Lesson 3 of 6

How do you write a rule for something that rises and falls again and again?

A tide, a wheel and a pendulum each repeat. This lesson asks how few measurements you need to write one rule for each of them, and where each measurement goes in that rule.

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 2: state the amplitude and the period of y = 3 sin 4x.
    AnswerAmplitude 3 and period 90°. 360 ÷ 4.
  2. From Year 10: the graph of y = f(x) is moved 5 units to the right. What is the new equation?
    Answery = f(x − 5). Every point moves right by 5.
  3. From Year 7: the highest value of a quantity is 10 and the lowest is 2. What is the value exactly halfway between them?
    Answer6. (10 + 2) ÷ 2.
Words used in this lesson
midline
the level halfway between the highest and lowest values. For example: the line y = 5 for a wave between 3 and 7.
amplitude
half the distance between the highest and lowest values. For example: 2 for a wave between 3 and 7.
period
the time or angle after which the pattern repeats. For example: 12 hours for a tide.
phase shift
a movement of a periodic graph to the right or left. For example: the 70° in sin (x − 70°).
model
an equation that describes a real situation. For example: the depth of water against time.

The harbour mouth

A harbour authority in Dar es Salaam records the depth of water at the harbour mouth. Twice a day it rises to a high tide and falls to a low tide, and the pattern repeats. A pilot wants a rule for the depth at any time, so that she can say when a deep-keeled ship may enter.

Predict first: of the following, which do you need to know to write a rule: the highest depth, the lowest depth, the time between two high tides, the time of one high tide, the colour of the ship? And how would you use each one in an equation of the form y = a sin b(x − c) + d?

0
0
0midline, y =
90x-value of the first maximum

y = 2 sin x: midline y = 0, highest value 2, lowest −2. The first maximum is at x = 90°.

Show me the steps
  1. c = 0, d = 0: this is y = 2 sin x. The midline is y = 0, the highest value is 2 and the first maximum is at x = 90°.
  2. Set c = 60°: the whole wave moves 60° to the right. The midline and the highest and lowest values are unchanged, but the first maximum is now at x = 90° + 60° = 150°.
  3. Set c = 0 and d = 2: the wave moves up by 2. The midline is y = 2, the highest value is 2 + 2 = 4 and the lowest is 2 − 2 = 0. The first maximum is still at x = 90°.

The family y = a sin b(x − c) + d describes anything that repeats smoothly.

  • d is the midline: the level halfway between the highest and lowest values, (highest + lowest) ÷ 2.
  • a is the amplitude: half the distance between them, (highest − lowest) ÷ 2.
  • b sets the period: 360° ÷ b, or 360 divided by the period for the number of waves.
  • c is the shift to the right, as in Year 10: replace x by x − c. The first maximum of a sine is at x = 90° ÷ b + c.
  • If the quantity starts at its highest or lowest value, use a cosine, or a negative cosine: cos x = sin (x + 90°).

The highest value is d + a, which is why d is the middle.

10
6height of the seat (m)
60angle turned (degrees)

t = 10 s: the wheel has turned 60° from the bottom and the seat is 6 m above the ground.

Show me the steps
  1. t = 0: the seat is at the bottom, 10 − 8 = 2 m. The wheel turns 6° every second, so 6t degrees in t seconds.
  2. t = 30 s: the wheel has turned 180° and the seat is at the top, 10 + 8 = 18 m. The graph has made half a wave.
  3. t = 45 s: the wheel has turned 270°. The seat is level with the axle, 10 m above the ground. At t = 60 s it is back at the bottom and the graph begins again.

Inside the bracket, the shift is the opposite of what the sign suggests. The bracket x − 70° moves the graph 70° to the right, not to the left. And the shift c is inside the bracket with b: in y = sin 3(x − 10°) the shift is 10°, but in y = sin (3x − 10°) it is 10° ÷ 3.

Worked examples

The highest tide at a port is 8.6 m and the lowest is 2.2 m. Find the midline and the amplitude.

  1. The midline is halfway: (8.6 + 2.2) ÷ 2 = 5.4 m.
  2. The amplitude is half the range: (8.6 − 2.2) ÷ 2 = 3.2 m.

The same tide repeats every 12 hours and the depth is highest at t = 0. Write the model and find the depth at t = 2.

  1. It starts at its highest value, so use a cosine: d = 5.4 + 3.2 cos bt. The period is 12 hours, so b = 360 ÷ 12 = 30.
  2. d = 5.4 + 3.2 cos(30t). At t = 2: cos 60° = 0.5, so d = 5.4 + 1.6 = 7.0 m.

Your go. The first step is given. A wheel has radius 6 m and its axle is 7 m above the ground. It turns once every 24 seconds, and a rider starts at the lowest point. Write a model for the rider's height h at time t.

  1. The rider starts at the lowest value, so the model is h = 7 − 6 cos bt: midline 7, amplitude 6, upside down.
  2. Use the time for one turn to find b, then write the model.
Check the answer

b = 360 ÷ 24 = 15, so h = 7 − 6 cos(15t). At t = 0 this is 7 − 6 = 1 m, the lowest point, and it is 13 m at t = 12 s, the top. A common slip is to write +6 cos, which starts at the top.

With a GDC (optional)

Graph the model with the angle mode set to degrees, and choose a window that shows two or three full periods: for a period of 12 hours, x from 0 to 36. Use the table of values with a step of 1 to compare with your hand calculation, and the maximum and minimum tools to check the highest and lowest values.

The GDC reads values off a model you have already written. Writing the model from the highest value, the lowest value and the period is the part to do by hand.

Check before you go on

1. Consider y = 3 sin (x − 40°) + 5. Which statement about its highest point is correct?

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. The height of a buoy above the seabed, in metres, is h = 12 + 3 sin(30t) at t hours. Find its greatest height.

3. At a port in Chittagong the highest tide is 9.2 m and the lowest is 3.6 m. Find the midline of a model of the depth, in metres, to 1 decimal place.

Level 2 · Building

4. The height in centimetres of the water in a wave tank in Busan is h = 20 + 6 sin(15(t − 4)). Find h when t = 10, to 1 decimal place.

5. A model is h = 10 + 4 sin(30(t − 2)). Find the first time t, with t ≥ 2, at which h is greatest.

Level 3 · Stretch

6. A wheel has radius 12 m with its axle 14 m above the ground and turns once every 40 s. A rider starts at the lowest point, so h = 14 − 12 cos(9t). Find the rider's height after 15 s, to 1 decimal place.

7. The depth of water in a harbour is d = 5 + 2 sin(30t) metres, t in hours. A boat needs at least 6 m. For how many hours, in the first 12 hours, is d at least 6?

Something to think about

The wheel model h = 10 − 8 cos(6t) can also be written with a sine, h = 10 + 8 sin(6(t − c)). By how many seconds, c, must the sine be moved to the right to match it?

One way to think about it

The sine 10 + 8 sin(6t) has its lowest point at 45 s, three quarters of the way through the period. The wheel's lowest point is at 0 s, which repeats at 60 s. Moving the sine right by 15 s puts its lowest point at 60 s, so c = 15 and h = 10 + 8 sin(6(t − 15)). At t = 0 this is 10 + 8 sin(−90°) = 2, the lowest point.

Exit ticket

8. A tide has a highest depth of 10 m and a lowest of 2 m. It repeats every 12 hours, and at t = 0 the depth is halfway between the two and rising. Which model is correct?