C. Trigonometric Functions · Sine Functions

Solving Problems with Sine Models

C3.3About 75 minutes

Learning goals

I can find the period, maximum, minimum and axis of a sine model with time as the input.
I can evaluate a sine model to answer a question in context.
I can find when a sine model reaches a given value, using a graph or by solving with sin⁻¹.
I can find how long a quantity stays above or below a value in each cycle.

Words to know

model input/output
The independent variable (often time) and the dependent variable (often height or depth) of a model.

intersect (technology)
A graphing calculator feature that finds where two graphs meet, such as a model and a horizontal line.

window
An interval of time during which a condition is true, such as 'deep enough to dock'.

Making the model work for you

A model is only useful if you can answer questions with it: How high is the seat after 10 s? When is the water deep enough for the ferry? How long each day is the temperature above freezing? Today you will answer questions like these in two ways: by reading a graph (one you are given or one you make with technology) and by working with the equation using your calculator.

Remember from Lessons 3 and 11

sin θ = k (with −1 < k < 1) has two solutions in each 360° cycle. If sin⁻¹(k) gives the angle α, the other is 180° − α (and then add or subtract 360° for more).
In a time model like h(t) = a sin(kt − d) + c, the expression inside is an angle in degrees, and the period is 360 ÷ k.

Strategy for solving problems with a sine model

Understand the model. Identify the max, min, axis, amplitude and period, and what the variables mean (units!).
To find an output ("How high at t = 10?"): substitute and evaluate. Check that your calculator is in degree mode.
To find inputs ("When is it 32 m high?"):

Graph method: draw the horizontal line at that output and read where it meets the curve.
Equation method: isolate the sine, use sin⁻¹ to get one angle, use 180° minus that angle for the second, then solve for the variable. Add one period at a time to find later times.

Answer in context, with units and sensible rounding, and check that the answer is inside the domain.

Worked example

Example 1: A Ferris wheel over time

A Ferris wheel seat's height is h(t) = 20 sin(6t − 90°) + 22, where h is in metres and t is the time in seconds after boarding. (a) Find the max, min and period. (b) How high is the seat at t = 10 s? (c) When, during the first turn, is the seat exactly 32 m high? (d) For how long during each turn is the seat above 32 m?

(a) Max = 22 + 20 = 42 m, min = 22 − 20 = 2 m. The angle 6t − 90° increases by 360° when t increases by 360 ÷ 6 = 60 s, so the period is 60 s (one turn per minute).
(b) h(10) = 20 sin(60° − 90°) + 22 = 20 sin(−30°) + 22 = 20(−0.5) + 22 = 12 m.
(c) 20 sin(6t − 90°) + 22 = 32 → 20 sin(6t − 90°) = 10 → sin(6t − 90°) = 0.5. In the first turn, 6t − 90° runs from −90° to 270°, and sin = 0.5 at 30° and 150°.
6t − 90° = 30° gives 6t = 120°, t = 20 s. 6t − 90° = 150° gives 6t = 240°, t = 40 s. Check: 6(40) − 90° = 150° and 20 sin 150° + 22 = 32 ✓.
(d) Between 20 s and 40 s the seat goes over the top (the max is at t = 30 s, where 6t − 90° = 90°). So it is above 32 m for 40 − 20 = 20 s of each 60 s turn.

Max 42 m, min 2 m, period 60 s; h(10) = 12 m; 32 m at t = 20 s and 40 s; above 32 m for 20 s per turn.

h(t) = 20 sin(6t − 90°) + 22 for two turns. The dashed line is h = 32. The seat is above 32 m in the shaded intervals: 20 s to 40 s and 80 s to 100 s.

Check every answer

A common slip is to forget to add back the shift. In Example 1(c), solving 6t − 90° = 150° as 6t = 150° would give t = 25 s, but h(25) ≈ 39.3 m, not 32 m. Always substitute your answers back into the equation, or compare them with the graph.

Worked example

Example 2: Enough water for the ferry

The water depth at a ferry dock is modelled by d(t) = 3 sin(30t) + 5, where d is in metres and t is hours after midnight. The ferry needs at least 6.5 m of water. During which times of the day can it dock?

Period = 360 ÷ 30 = 12 h; max 8 m, min 2 m.
Solve d(t) = 6.5: 3 sin(30t) + 5 = 6.5 → sin(30t) = 0.5.
30t = 30° or 30t = 150° (first cycle). So t = 1 h or t = 5 h.
Between these times the sine is above 0.5 (the max is at 30t = 90°, t = 3 h), so the depth is at least 6.5 m from 1:00 a.m. to 5:00 a.m.
Add one period (12 h) for the next window: 13 h to 17 h, that is, 1:00 p.m. to 5:00 p.m.

The ferry can dock from 1:00 a.m. to 5:00 a.m. and from 1:00 p.m. to 5:00 p.m.

d(t) = 3 sin(30t) + 5 over one day. The depth is at least 6.5 m in the two shaded windows.

Worked example

Example 3: Working from a given graph only

Use the tide graph above (without the equation). (a) What is the depth at 9:00 a.m.? (b) At what times between noon and midnight is the depth exactly 5 m? (c) What is the lowest depth, and when does it first happen?

(a) At t = 9, the curve is at its lowest point: 2 m.
(b) The curve crosses its axis (5 m) every half period: t = 0, 6, 12, 18, 24. Between noon (t = 12) and midnight (t = 24): 12:00 noon, 6:00 p.m. and midnight.
(c) The lowest point is 2 m, first at t = 9 h, 9:00 a.m.

(a) 2 m (b) noon, 6:00 p.m., midnight (c) 2 m, first at 9:00 a.m.

Using technology

Enter the model in a graphing calculator or app (degree mode), then use the intersect feature with a horizontal line such as y = 32, or trace to read values. Set the window to match the context: for Example 2, 0 ≤ x ≤ 24 and 0 ≤ y ≤ 9.

Just for fun

Why did the ferry captain love sine models?

Because whenever the tide went out, she knew it would come back with a positive attitude.

Hands-on activity

Plan your outdoor day with a tide model

You need: paper, a pencil, a calculator, and (optional) a graphing app.

A beach on the Bay of Fundy has tide depth (above the lowest point of the beach) modelled by d(t) = 6 sin(30t − 60°) + 6, where t is hours after 6:00 a.m. and d is in metres. The tide flats are safe to walk on only when d ≤ 1.5 m (follow local signs and guides at a real beach; tides there come in fast).

Make a table of d(t) for t = 0, 1, 2, ..., 12 and plot it.
Find the period, max and min.
Solve d(t) = 1.5 by hand. Give the times as clock times.
Shade the times on your graph when walking is allowed. Plan a 2-hour walk that fits.

What to look for: period 12 h, max 12 m, min 0 m (at t = 11, 5:00 p.m.). d = 1.5 means sin(30t − 60°) = −0.75, so 30t − 60° ≈ 228.6° or 311.4°, giving t ≈ 9.6 h and 12.4 h: about 3:37 p.m. to 6:23 p.m. The equation also gives t ≈ 0.4 h: the flats are already walkable at 6:00 a.m. (d(0) ≈ 0.8 m) until about 6:23 a.m., the end of the previous low-tide window. A 2-hour walk from about 4:00 p.m. to 6:00 p.m. fits.

Your turn

For h(t) = 20 sin(6t − 90°) + 22, find the height at t = 25 s.

For the same wheel, when in the first turn is the seat 12 m high?

For d(t) = 3 sin(30t) + 5, find the depth at 7:00 a.m.

For d(t) = 3 sin(30t) + 5, when between midnight and noon is the water 8 m deep?

A model is T(m) = 14 sin(30m − 120°) + 6, the average temperature in °C in month m (January = 1). Find the warmest month.

For T(m) above, which months have an average temperature below 0 °C? Evaluate T for the months near winter.

More practice: warm-up

Period of h(t) = 5 sin(12t) + 9?

Max of h(t) = 5 sin(12t) + 9?

Evaluate 4 sin(30°) + 2.

sin θ = 0.5 for which two angles between 0° and 360°?

sin θ = −1 for which angle between 0° and 360°?

More practice: on your own

A pendulum's horizontal position is x(t) = 15 sin(180t), in cm, t in seconds. Find the period, and the position at t = 0.25 s and t = 1.5 s.

A wheel: h(t) = 10 sin(9t − 90°) + 12 (t in s). How long does one turn take, and when is the seat first at the top?

For the wheel above, for how long in each turn is the seat below 7 m?

Using only the graph of Example 2, estimate how many hours each day the water is deeper than 6.5 m.

Pose a question about the Ferris wheel in Example 1 that can be answered with the model, and answer it.

Challenge

For h(t) = 20 sin(6t − 90°) + 22, find all times in the first 2 minutes when the seat is 37 m high (nearest 0.1 s).

A tide model is d(t) = 2.5 sin(30t − 90°) + 3.5 (t hours after midnight). A sailboat needs 4.75 m. Find the first time window after midnight.

Write your own problem using the model d(t) = 3 sin(30t) + 5 that needs the period to answer, then solve it.

Check your understanding

Quiz

h(t) = 20 sin(6t − 90°) + 22. What is the period?

d(t) = 3 sin(30t) + 5. What is the depth at t = 3?

Solving 4 sin θ + 1 = 3 for 0° ≤ θ ≤ 360° gives

A model has max 42 m and the seat is above 32 m from t = 20 s to t = 40 s each 60 s turn. What fraction of each turn is above 32 m?

After solving a sine equation, what is the best way to catch a mistake?

d(t) = 3 sin(30t) + 5 is at least 6.5 m from t = 1 to t = 5. When is the next window?

For parents and teachers

Big idea: Posing and solving problems with a sine model, using a given graph or a graph made with technology from an equation (C3.3).

How to help: Have your teen explain what each number in a model means in the situation ("the 22 is the height of the centre") before calculating. Ask them to check every answer by substituting it back.

Common difficulties:

Finding only one solution of sin θ = k. Remember the second angle, 180° minus the first, and later cycles.
Forgetting to undo the steps in the right order (subtract c, divide by a, then use sin⁻¹, then solve for t).
Degree/radian mode errors, and giving answers outside the domain (such as before the ride starts).