Lesson 7 — Trigonometric Functions as Waves

Summary

Learners meet sine and cosine as periodic functions — functions that repeat — and name the four features that shape a wave: amplitude, period/frequency, midline, and phase shift. They read each feature from both an equation (y = A·sin(B(x − C)) + D) and a graph, and meet the Indian origin of the word “sine.”

Objectives

• Model periodic phenomena with sine and cosine, and interpret amplitude, frequency (period), and shift in an equation and a graph. (D05.S2.11.01)

Connection

Many things in the world go around and come back: your heartbeat, the tides, the seasons, a swing, a sound’s vibration, the daylight over a year. All of them repeat — they are periodic. The sine and cosine functions are the language for that repetition: one shape that rises, falls, and returns, which you can stretch taller, squeeze faster, slide up, or shift along to match whatever is repeating in front of you.

Materials

• Wave problem sheet
• Graph paper
• Math journal

Preparation

• Copy or draw the problem sheet.
• Retrieval: from Grade 10, right-triangle sine and cosine as ratios (D05.S3.10.01). Today they become functions of an angle that keep on repeating.
• Prepare worked examples for reading amplitude, period, midline, and shift.

Facilitator note

This lesson is written to the learner (“you”). The ideas to land: in y = A·sin(B(x − C)) + D, A is amplitude (height from midline), the period is 2π/B (and frequency is its reciprocal), C is phase shift (left/right), and D is the midline (up/down). Teach the four features one at a time, each with a graph, then combine them (S-011).

The intellectual lens: four numbers (A, B, C, D) control a whole repeating shape — a small alphabet for a huge family of waves. The global lens: the word “sine” comes from a mistranslation of the Sanskrit jya (chord/half-chord) through Arabic jiba to Latin sinus — trigonometry grew in India and the Islamic world, not one place (S-240, S-429). The critical-thinking lens: learners check each feature by marking it on the graph. The environment lens: day length, tides, and seasonal temperature are real periodic signals — reading their amplitude and frequency is reading the planet’s rhythm. Preview: Lesson 8 fits these features to real data.

Procedure

1. Recall (5 min). From Grade 10, what is sin θ in a right triangle? Today θ keeps growing, and the ratio becomes a repeating wave.
2. Meet the wave (10 min). Plot y = sin x. It rises to 1, falls to −1, and repeats every 2π. That is a periodic function. The midline is the center line (here y = 0); amplitude is the height from midline to peak (here 1); the period is the length of one full cycle (here 2π); frequency is how many cycles fit in a unit of time (1/period).
3. The four features (15 min). In y = A·sin(B(x − C)) + D:
   • A (amplitude): y = 3·sin x is three times as tall (amplitude 3).
   • B (period): y = sin(2x) finishes a cycle in π, not 2π — period 2π/2 = π, so it is twice as frequent.
   • C (phase shift): y = sin(x − π/2) is shifted right by π/2.
   • D (midline): y = sin x + 2 rides on the line y = 2. Worked example: y = 2·sin(3(x − π/4)) + 1 has amplitude 2, period 2π/3, phase shift π/4 right, midline y = 1.
4. Guided practice (12 min). With a partner, read each feature from: (a) y = 4·sin(2x) − 1 (b) y = sin(x/2) + 3 (c) a drawn wave whose peaks are at 5 and −1 and which repeats every 4 units. Mark amplitude, midline, and period on each.
5. Independent practice (5 min). In your journal, sketch one wave and label its amplitude, midline, and period.
6. Close (8 min). In your own words, say what each of A, B, C, D does, and name one real thing that repeats.

Differentiation

• Support: Change only one feature at a time (amplitude alone, then midline alone) and trace the wave with a finger.
• Extension: Write the equation of a wave from a graph given its peak, trough, and period, including a phase shift.

Assessment

• Formative (peer + self): Can the learner state the amplitude, period/frequency, midline, and phase shift of a sine or cosine function from an equation and a graph?
• Portfolio artifact (unit): The labeled wave sketch, added to the pattern toolkit.

Home connection

Find one repeating thing at home — a ceiling fan’s spin, a pendulum, a dripping tap at steady intervals — and describe its “amplitude” and “frequency” in your own words.

Resources

• On worked examples and guided practice: Kirschner, Sweller & Clark (2006), https://doi.org/10.1207/s15326985ep4102_1 (S-011).
• On the Indian origin of the word “sine” and trigonometry’s global history: MacTutor History of Mathematics Archive (S-240); Boyer & Merzbach, A History of Mathematics (S-429); OpenStax, Algebra and Trigonometry (S-435).