Pose and solve real-world sine-function problems

Ontario Grade 11 Mathematics

Trigonometric Functions

Ontario Grade 11 Mathematics — C3.3

A point on a steadily turning ride rises and falls as the ride turns. Its height repeats after each full turn. A sine function can model this kind of regular change. In this lesson, you will connect a situation to a table, a graph, and an equation. You will also learn to ask a useful question about a situation and answer it with the model. The model used here begins at its middle value and rises.

What you will learn

Recognize real situations that repeat in a regular cycle.
Use a sine function to model a quantity that rises and falls over time.
Find the midline, amplitude, and period from a situation or its graph.
Pose a question about a real situation, solve it with a sine model, and check the answer.

1. Bridge: read the situation and its graph

A function connects an input to an output. In a height model, time is the input and height is the output. Give both quantities units. For example, time could be in seconds and height in metres.

A cycle is one complete repeat of a pattern. The period is the amount of input needed for one complete cycle. If a ride returns to the same position every 24 seconds, its period is 24 seconds.

The maximum is the greatest output in a cycle. The minimum is the least output. The midline is halfway between them. The amplitude is the distance from the midline to either extreme. These values describe how high and low the pattern goes.

M=\frac{y_{\max}+y_{\min}}{2},\quad A=\frac{y_{\max}-y_{\min}}{2}

The period measures the length of one cycle.
The midline is halfway between the maximum and minimum.
The amplitude is half the difference between the maximum and minimum.

2. Describe a repeating pattern with a sine model

A sine graph has a smooth, repeating rise and fall. In the model used here, the graph begins at the midline, rises to a maximum, falls through the midline to a minimum, and returns to the midline.

The equation

h(t)=A\sin(kt)+M

describes this starting pattern when

A

and

k

are positive. The letter

h

represents the output, such as height. The input

t

could represent time. The amplitude is

A

, and the midline value is

M

. The number

k

controls how quickly the pattern repeats.

This equation uses degrees for the sine input. One full cycle corresponds to

360^\circ

. If the period is

P

, divide

360^\circ

by

P

to find

k

. Keep time units consistent: if

P

is in seconds, enter time in seconds too. Set a calculator to degree mode when evaluating this model.

A sine model is suitable when the real pattern repeats in a reasonably regular way and matches the model’s starting behaviour. Before using it, check that the situation starts at the midline and initially rises.

h(t)=A\sin(kt)+M,\quad k=\frac{360^\circ}{P}

Use this model when the pattern starts at the midline and rises.
Use the same input units for the period and the time values.
Check the starting behaviour and degree setting before calculating.

3. Connect a description, table, graph, and equation

A description explains what is happening and gives units. A table lists selected input-output pairs. A graph shows the whole repeating shape. An equation lets you calculate an output for a chosen input. These are different ways to describe the same situation.

For a pattern that starts at the midline and rises, a quarter of a cycle later it reaches its maximum. Half a cycle after the start, it crosses the midline while falling. Three quarters of a cycle after the start, it reaches its minimum. At one full period, it returns to its starting value.

The maximum and minimum can be checked from the model. The maximum is the midline plus the amplitude. The minimum is the midline minus the amplitude. A graphing tool can help show the curve, but check its labels, units, scale, and degree setting.

h_{\max}=M+A,\quad h_{\min}=M-A

One full repeat along the horizontal axis shows the period.
The midline marks the centre of the graph’s highest and lowest values.
Use the maximum and minimum to check a graph or calculated answer.

4. Pose a question, solve it, and check

A real-world problem might ask for the output at a given time. You can also pose a question by choosing a useful time or target output from the situation. State what the variables mean, what information is known, and what you want to find.

Find the midline, amplitude, and period from the situation, table, or graph. Build a model that matches the starting behaviour. To find an output, substitute the given input and calculate. If you need a time for a target output, use a graph or a calculator’s equation-solving feature. Check that the time belongs to the interval asked about.

Finish by stating the answer in context, with units. A predicted output should be between the model’s maximum and minimum. A time after the start should not be negative. If an answer does not make sense, check the model, units, and calculator setting.

Identify the unknown and state what each variable represents.
Include units and answer in the context of the situation.
Check that the result is within the model’s possible range.

Key points in one cycle of the ride model

Time	Position in cycle	Height
0 s	Midline, rising	14 m
6 s	Maximum	20 m
12 s	Midline, falling	14 m
18 s	Minimum	8 m
24 s	Back at start	14 m

Worked example

Height of a platform on a rotating ride

A platform moves steadily around a ride. Its lowest height is 8 metres and its greatest height is 20 metres. At time zero, it is at the midline and moving upward. One full rotation takes 24 seconds. Pose and answer this question: How high is the platform 5 seconds after the start?

Find the centre and range
The midline is halfway between the lowest and greatest heights. The amplitude is half the difference between those heights. These values describe the centre and size of the platform’s repeating movement.
$M=\frac{20+8}{2}=14,\quad A=\frac{20-8}{2}=6$
Find the cycle rate
The period is 24 seconds. Divide the full turn of $360^\circ$ by 24 seconds to find how many degrees the sine input changes each second.
$k=\frac{360^\circ}{24}=15^\circ/\mathrm{s}$
Write a model
The platform starts at the midline and rises, so the model form fits. Height is measured in metres and time in seconds.
$h(t)=6\sin(15t)+14$
Calculate the height
Substitute 5 seconds for $t$ . In degree mode, the sine input is $75^\circ$ . The sine value is about 0.966, so the platform is about 19.8 metres high.
$h(5)=6\sin(75^\circ)+14\approx19.8$

Answer: The platform is about 19.8 metres high 5 seconds after the start.

Check: The model’s lowest and greatest heights are 8 metres and 20 metres, so 19.8 metres is possible. One quarter of the 24-second period is 6 seconds, when the platform reaches its greatest height. Since 5 seconds is close to 6 seconds, a height close to 20 metres makes sense.

Common mistakes and how to avoid them

Calling the maximum the amplitude.

Correction: Find the midline first. The amplitude is the distance from the midline to the maximum or minimum.

Using different time units for the period and the input.

Correction: Use consistent units. For example, convert minutes to seconds if the period is given in seconds.

Leaving a calculator in a setting that does not match the degree-based model.

Correction: Set the calculator to degree mode before evaluating the sine input.

Choosing a model that starts at the wrong height or moves in the wrong direction.

Correction: Check the starting value and whether the quantity first rises or falls. The model in this lesson begins at the midline and rises.

Reporting a number without units or checking its size.

Correction: Include units and compare the result with the model’s minimum and maximum.

Lesson summary

A sine function can model a regular repeating rise and fall.
Use the maximum and minimum to find the midline and amplitude.
For a model that starts at the midline and rises, use $h(t)=A\sin(kt)+M$ and find $k$ by dividing $360^\circ$ by the period.
Keep units consistent, use degree mode, and check that the answer fits the situation.

Check your understanding

Question 1

A quantity has a maximum of 18 units and a minimum of 6 units. What are its midline and amplitude?

Midline 12; amplitude 6
Midline 6; amplitude 12
Midline 12; amplitude 12
Midline 24; amplitude 6

Show answer and explanation

Midline 12; amplitude 6

The midline is halfway between 18 and 6, which is 12. The amplitude is half their difference, which is 6.

Question 2

A simple sine model has a period of 30 seconds and uses degree input. What value of

k

should it use?

$12^\circ/\mathrm{s}$
$30^\circ/\mathrm{s}$
$360^\circ/\mathrm{s}$
$390^\circ/\mathrm{s}$

Show answer and explanation

12^\circ/\mathrm{s}

Divide

360^\circ

by 30 seconds. This gives

12^\circ

per second.

Question 3

A measured quantity begins at its midline and initially rises. Which model form from this lesson matches that behaviour?

$h(t)=A\sin(kt)+M$
$h(t)=A\sin(kt)-M$
$h(t)=M\sin(kt)+A$
$h(t)=A\sin(t)+M$ for every possible period

Show answer and explanation

h(t)=A\sin(kt)+M

At time zero, the sine value is zero, so this form starts at

M

. With positive

A

and

k

, it initially rises. The period determines the value of

k

.

Key terms

Cycle: One complete repeat of a pattern.
Period: The input length needed for one complete cycle.
Midline: The value halfway between the maximum and minimum.
Amplitude: The distance from the midline to a maximum or minimum.
Sine function: A function with a smooth, repeating rise-and-fall pattern.

Continue through MCF3M

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Mathematics (MCF3M), expectation C3.3. 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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