B2.7 Solve sinusoidal applications in radians | Ontario Grade 12 Mathematics

Ontario Grade 12 Mathematics

Trigonometric Functions

Build a model, connect it to a situation, and find the times or values that answer a question.

Sinusoidal models describe quantities that repeat in a smooth cycle. Examples include the height of a point on a rotating wheel and the position of a swinging object over time. In an application, the model connects a changing quantity to an input such as time. This lesson focuses on solving those applications when angles are measured in radians. Before calculating, identify the cycle, set up a model that fits the situation, and keep track of which solutions are meaningful.

What you will learn

Identify the midline, amplitude, and period in a sinusoidal situation.
Choose and use a sine or cosine model that matches the starting conditions.
Use radians to calculate unknown values or times in a sinusoidal application.
Check that solutions make sense in the context and within the requested interval.

1. Prerequisite bridge: read a repeating cycle

A sinusoidal graph is a smooth wave that repeats. Its highest and lowest values are called the maximum and minimum. The midline is the horizontal line halfway between them. The amplitude is the vertical distance from the midline to a maximum or minimum. It is always positive.

The period is the input distance needed for one complete cycle. For a time-based model, it tells how long one cycle takes. A cycle may be described using sine or cosine. Sine begins at the midline when its angle is zero. Cosine begins at a maximum when its angle is zero. Reflections or shifts can change these starting positions.

Radians measure angles. One complete turn is

2\pi

radians, and a half-turn is

\pi

radians. A calculator must be in radian mode when you evaluate trigonometric expressions or use inverse trigonometric functions in a radian model.

\text{midline}=\frac{\text{maximum}+\text{minimum}}{2},\quad \text{amplitude}=\frac{\text{maximum}-\text{minimum}}{2}

Midline: average of the maximum and minimum.
Amplitude: half the difference between the maximum and minimum.
Period: input change for one complete cycle.
Use radian mode for calculations involving radian angles.

2. From a situation to a sinusoidal model

Begin with the setting. Decide what the input and output represent, and write their units. For example, the input might be time in seconds and the output might be height in metres. Identify the maximum and minimum output values. These give the midline and amplitude.

Next, determine the period from the situation. A model with period

P

has an angle that completes one full turn, or

2\pi

radians, during that input interval. Therefore, the multiplier on the input is

2\pi/P

. This multiplier is often called the angular frequency; here it simply converts the input into the angle used by sine or cosine.

Choose sine or cosine to match a known point in the cycle. If the quantity starts at a maximum, a cosine model is often convenient. If it starts at the midline and rises, a sine model is often convenient. A negative sign can represent a reflected curve. A horizontal shift can place a maximum, minimum, or midline crossing at the correct input value.

A common model is

y=A\sin(B(x-D))+C

or

y=A\cos(B(x-D))+C

. In either form,

C

is the midline, |A| is the amplitude, and the period is

2\pi/|B|

. The value

D

shifts the cycle horizontally. These symbols describe features of the situation, so their units should be interpreted in context.

P=\frac{2\pi}{|B|}

Use the maximum and minimum to find the midline and amplitude.
Use the period to determine the input-to-angle multiplier.
Select a model that matches a known starting value and direction.
Check that the model’s units and starting behaviour fit the situation.

3. Solve and interpret the model

An application question may ask for an output at a given time or for the time when an output reaches a specified value. For an output question, substitute the input into the model and evaluate. For a time question, set the model equal to the target output and solve the trigonometric equation.

A target output can occur more than once in a cycle. Sine and cosine repeat, so solving for one angle is not always enough. Use the graph, a cycle sketch, or the situation to find every solution in the requested interval. Then convert angles to input values using the model’s angle multiplier and shift.

Inverse sine and inverse cosine on a calculator usually return one angle, not every angle in the cycle. Use the symmetry of the graph to identify any other angle that fits. Finally, check each answer by substituting it into the model and confirm that it falls within the time interval and makes sense in the situation.

\theta=B(x-D)

A target level can be reached multiple times in one cycle.
Use the requested interval and the graph’s shape to select all relevant solutions.
Convert angles to input values and include units.
Check answers in the original model.

4. Application habits

A good solution tells the reader what the variables mean. State the units for the input and output. If the answer is a time, report it as a time in the situation, not just as an angle. Rounding should suit the context. For example, a time measured in seconds might be reported to the nearest hundredth of a second if the model and question support that precision.

It is useful to check the model at important points in the cycle. At a maximum, the output should equal the midline plus the amplitude. At a minimum, it should equal the midline minus the amplitude. After one period, the model should return to the same value. These checks can reveal a wrong sign, period, or shift before you solve the full question.

Name variables and units clearly.
Report values in the form requested by the application.
Use maximum, minimum, and period checks to test the model.

Wheel heights at key points in one rotation

Time (s)	Angle (rad)	Height (m)
0	0	1
3	$\frac{\pi}{2}$	4
6	\pi	7
9	$\frac{3\pi}{2}$	4
12	2\pi	1

Worked example

Height of a point on a rotating wheel

A point on a wheel moves in a vertical circle of radius

3

m. The wheel’s centre is

4

m above the ground. At time

t=0

, the point is at the bottom and begins moving upward. The wheel turns at

\frac{\pi}{6}

radians per second. Find the first two positive times when the point is

5

m above the ground.

Identify the cycle features
The lowest height is $4-3=1$ m and the highest is $4+3=7$ m. Thus the midline is $4$ m and the amplitude is $3$ m. One full turn takes $2\pi$ divided by the angular speed, so the period is $12$ seconds.
$P=\frac{2\pi}{\pi/6}=12$
Choose a model
The point starts at its minimum height, so a negative cosine model is convenient. The centre height is the midline, and the given turning rate is the angle multiplier. This model starts at $1$ m and then increases, as described.
$h(t)=4-3\cos\left(\frac{\pi}{6}t\right)$
Set the height to the target
Substitute the target height of $5$ m into the model. Rearranging gives a cosine value of $-\frac{1}{3}$ . Use the calculator in radian mode to find the angle in the first half of the cycle.
$5=4-3\cos\left(\frac{\pi}{6}t\right),\qquad \cos\left(\frac{\pi}{6}t\right)=-\frac{1}{3}$
Find both angles in one turn
The first angle is approximately $1.911$ radians. Cosine has the same value again later in the turn, at $2\pi-1.911\approx4.373$ radians. Both angles lie between $0$ and $2\pi$ , so they give the two crossings in the first rotation.
$\theta_1\approx1.911,\qquad \theta_2\approx2\pi-1.911=4.373$
Convert angles to times
The model’s angle is $\frac{\pi}{6}t$ , so divide each angle by $\frac{\pi}{6}$ . Round to the nearest hundredth. Both times are positive and are less than the $12$ -second period.
$t_1\approx3.65\text{ s},\qquad t_2\approx8.35\text{ s}$

Answer: The point is

5

m above the ground at approximately

3.65

s and

8.35

s after it starts moving.

Check: The point rises from its minimum at time zero, reaches its maximum halfway through the rotation at

6

s, and returns to the bottom at

12

s. The two calculated times fall on either side of

6

s, as expected for the same height.

Common mistakes and how to avoid them

Using the maximum as the midline or using the full maximum-to-minimum distance as the amplitude.

Correction: The midline is the average of the maximum and minimum. The amplitude is half their difference.

Using a period multiplier that does not match the stated period.

Correction: For a model with input multiplier

B

, check that its period is

2\pi/|B|

.

Leaving a calculator in degree mode for a radian model.

Correction: Set the calculator to radian mode before evaluating trigonometric or inverse trigonometric expressions.

Reporting only the first time a target value occurs.

Correction: Check the full requested interval. A sinusoidal quantity may reach the same target more than once in a cycle.

Reporting an angle as the answer when the question asks for time.

Correction: Convert the angle back to the input using the model, then report the time with units.

Lesson summary

Use the maximum and minimum to find the midline and amplitude.
Use the period to connect the input to an angle measured in radians.
Choose a sine or cosine model that matches the situation’s starting point and direction.
When solving for an input, look for every valid crossing in the requested interval.
Check the result in the model and report it with appropriate units.

Check your understanding

Question 1

A sinusoidal model has maximum value

11

and minimum value

3

. What are its midline and amplitude?

Midline $7$ , amplitude $4$
Midline $8$ , amplitude $4$
Midline $7$ , amplitude $8$
Midline $4$ , amplitude $7$

Show answer and explanation

Midline

7

, amplitude

4

The midline is

(11+3)/2=7

. The amplitude is

(11-3)/2=4

.

Question 2

A model is

y=2\sin(\frac{\pi}{4}t)+6

. What is its period?

$4$
$8$
$\frac{\pi}{4}$
$2\pi$

Show answer and explanation

8

Here

B=\frac{\pi}{4}

. The period is

2\pi/(\pi/4)=8

.

Question 3

A rotating point is modelled by

h(t)=5+2\cos(\frac{\pi}{3}t)

. What is its height at

t=3

seconds?

$3$
$5$
$7$
$0$

Show answer and explanation

3

At

t=3

, the angle is

\pi

. Since

\cos(\pi)=-1

, the height is

5+2(-1)=3

.

Key terms

Sinusoidal model: A sine- or cosine-based function used to represent a repeating, wave-shaped pattern.
Midline: The horizontal level halfway between a sinusoidal model’s maximum and minimum.
Amplitude: The vertical distance from the midline to a maximum or minimum.
Period: The input change needed for one complete repetition.
Radian: A unit for measuring angle; a full turn is $2\pi$ radians.
Angular frequency: The number that converts the input into the angle used by a sinusoidal model.

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Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 12 Mathematics (MHF4U), expectation B2.7. It is a study resource, not an official curriculum publication.

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B2.7 · Solve sinusoidal applications in radians

Ontario Grade 12 Mathematics

Build a model, connect it to a situation, and find the times or values that answer a question.

What you will learn

1. Prerequisite bridge: read a repeating cycle

2. From a situation to a sinusoidal model

3. Solve and interpret the model

4. Application habits

Wheel heights at key points in one rotation

Height of a point on a rotating wheel

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through MHF4U

About this lesson