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C. Trigonometric Functions · Sine Functions

Sine Functions: Unit Review

C2.1 C2.2 C2.3 C2.4 C2.5 C2.6 C2.7 C3.1 C3.2 C3.3About 40 minutes
Learning goals
  • I can review periodic functions, the sine function and its transformations.
  • I can find which lessons I need to revisit.

Words to know

sinusoidal model
An equation of the form y = a sin(x − d) + c (or with time as the input) that describes a periodic situation.

Let's review!

In this unit you described periodic functions and used the period to predict. You extended the sine ratio to every angle with the circle and built the sine function and its properties. You transformed the sine graph with a, d and c, linked those changes to real situations, and finally collected data, built sine models and used them to solve and pose problems.

There is one practice set for each lesson below. Try them without looking back first, then check your answers.

How to use this review

Do two or three sets at a time. Sketch a graph for every question about a sine function, even when it is not asked for. Keep your calculator in degree mode (sin 90 = 1). Use the five key points for every sketch, and check every equation by substituting a maximum or minimum point. If you miss two or more questions in a set, go back to that lesson (see the notes for parents).

Periodic functionsLessons 1–2Sine of any angleLessons 3–5Transforming sin xLessons 6–10Data and modelsLessons 11–14
The four parts of the unit.
xy0°90°180°270°360°−101234y = 2 sin(x − 90°) + 2y = sin x
A combined transformation from the review: y = 2 sin(x − 90°) + 2 (solid), with its key points, compared with y = sin x (dashed).
Just for fun

You now know how to predict tides, design Ferris wheels and describe the electricity in your walls, all with one wavy function.

Not bad for something that just keeps going up and down.

Periodic Functions: Cycle, Period, Axis and Amplitude

1

A periodic graph has max 11 and min 3. Find the axis and amplitude.

2

Peaks occur at t = 4 s and t = 10 s (with none in between). What is the period?

3

f has period 9 and f(2) = 5. Find f(29).

Reading Periodic Data and Making Predictions

1

High tide is at 5:15 a.m.; the period is 12.4 h. When is the next high tide? The next low tide?

2

Monthly gas use has max 280 m³ and min 40 m³. Find the amplitude and axis.

3

Why is predicting a value far in the future less reliable for weather-related data than for daylight?

Angles of Rotation and the Sine Ratio from 0° to 360°

1

P(−5, −12) is on the terminal arm of θ. Find sin θ and θ (0° ≤ θ ≤ 360°).

2

Find two angles between 0° and 360° with sin θ = 0.25 (nearest tenth).

3

In which quadrants is sin θ positive?

Graphing the Sine Function

1

Find sin 480° without a calculator.

2

Explain why y = sin x is a function even though sin 30° = sin 150°.

3

How many solutions does sin x = −0.6 have from 0° to 1080°?

Key Properties of f(x) = sin x

1

State the domain and range of y = sin x.

2

Where is y = sin x decreasing for 0° ≤ x ≤ 360°?

3

List the zeros of y = sin x for −360° ≤ x ≤ 0°.

Vertical Translations: f(x) = sin x + c

1

State the axis and range of y = sin x − 6.

2

A curve with amplitude 1 and period 360° has min 2.5. Write it in the form y = sin x + c.

3

Image of (270°, −1) under y = sin x + 4?

Vertical Stretches, Compressions and Reflections: f(x) = a sin x

1

Describe y = −3 sin x as transformations of y = sin x.

2

Where is the minimum of y = −3 sin x between 0° and 360°?

3

Write an equation for a graph with axis y = 0, amplitude 0.4 and a maximum at (90°, 0.4).

Horizontal Translations: f(x) = sin(x − d)

1

Which way and how far is y = sin(x + 75°) shifted?

2

Where is the first maximum to the right of the y-axis for y = sin(x − 50°)?

3

Find the y-intercept of y = sin(x + 30°).

Combining Transformations and Finding Equations

1

List the key points of one cycle of y = 2 sin(x − 90°) + 2.

2

State the range of y = 2 sin(x − 90°) + 2.

3

A graph with period 360° has max 9 at x = 120° and min 1. Write its equation (a > 0).

Real-World Changes and Graph Transformations

1

A Ferris wheel is moved to a hill 12 m higher. Which parameter changes, and how?

2

A swimmer swims the same pool more slowly. What happens to the period of the distance graph?

3

h = 9 sin(θ − 90°) + 11. Write the model for a seat that starts at the top of the same wheel.

Periodic or Sinusoidal? Tables, Graphs, Equations and Restrictions

1

Classify: (a) y = 5 sin(x − 10°) + 1 (b) y = 3ˣ (c) a repeating zigzag.

2

Find the period of h(t) = 4 sin(15t) + 8.

3

A 5-turn Ferris wheel ride: domain in degrees turned?

Collecting and Graphing Sinusoidal Data

1

Wheel data: max 74 cm, min 6 cm, mark starts on the axis going up. Write a model in degrees turned.

2

Monthly data with period 12 months: how many degrees per month in the model θ = 30m?

3

Give one source of secondary sinusoidal data.

Solving Problems with Sine Models

1

d(t) = 2 sin(30t) + 6 (t hours after midnight). Depth at 4:00 a.m.?

2

For the same model, when between midnight and noon is the depth 7 m?

3

For how long in each 12 h cycle is the depth above 7 m?

Modelling Projects: Posing and Solving Sine Problems

1

A wheel boards at 3 m and reaches 43 m. Write its model (boarding at the bottom).

2

Classify the question "For how many months is daylight above 14 h?"

3

V(t) = 170 sin(21 600t). Voltage at t = 1/80 s?

Check your understanding

Quiz

1.

A periodic function has max 12 and min −4. What are its amplitude and axis?

2.

P(8, −6) is on the terminal arm of θ. What is sin θ?

3.

Which is the range of y = 3 sin x?

4.

How is y = sin(x − 45°) related to y = sin x?

5.

Which equation has axis y = −2 and amplitude 5?

6.

Where is the maximum of y = −4 sin x + 1 between 0° and 360°?

7.

A graph with period 360° has max 7 at x = 60° and min −1. Its equation (a > 0) is

8.

A bigger Ferris wheel has the same boarding height. Which parameters of h = a sin(θ − 90°) + c change?

9.

h(t) = 6 sin(20t) + 10. What is the period?

10.

d(t) = 4 sin(30t) + 5. At which time between t = 0 and t = 12 is the depth exactly 9 m?

For parents and teachers

Use the review to find what to revisit:

  • Mixing up period and amplitude, or using the whole max-to-min distance as the amplitude → Lesson 1 (Periodic Functions).
  • Errors converting decimal hours, or adding a whole period instead of half → Lesson 2 (Reading Periodic Data and Making Predictions).
  • Wrong signs for sin θ, or finding only one angle with a given sine → Lesson 3 (Angles of Rotation and the Sine Ratio).
  • Calculator in the wrong mode, or confusion about why sin x is a function → Lesson 4 (Graphing the Sine Function).
  • Errors with domain, range, zeros or increasing/decreasing intervals → Lesson 5 (Key Properties).
  • Moving the graph the wrong way: up/down errors → Lesson 6; stretch, compression and reflection errors (negative amplitude, max at the wrong place) → Lesson 7; left/right sign errors → Lesson 8.
  • Wrong order of transformations, or equations from graphs that do not check → Lesson 9 (Combining Transformations).
  • Not linking a change in a situation to the right parameter → Lesson 10 (Real-World Changes).
  • Calling a zigzag sinusoidal, or ignoring context restrictions → Lesson 11 (Periodic or Sinusoidal?).
  • Trouble turning data into a model → Lesson 12 (Collecting and Graphing Sinusoidal Data).
  • Finding only one time when a value occurs, or not adding the period → Lesson 13 (Solving Problems).
  • Difficulty writing or answering their own questions → Lesson 14 (Modelling Projects).

Encouragement: This unit combines a lot of earlier ideas (functions, transformations, trigonometry). Ask your teen to explain one model in their own words: what each number means and how they would check it.