Mathematics · Secondary 4 · Geometry and Trigonometry · Semester 1

Graphs of Trigonometric Functions

Students will sketch and interpret the graphs of sine, cosine, and tangent functions for angles between 0° and 360°.

• MOE Syllabus Outcomes
• MOE: Trigonometry - S4

About This Topic

Graphs of trigonometric functions introduce students to periodic behaviour through sine, cosine, and tangent from 0° to 360°. Sine and cosine share a period of 360° and amplitude of 1, while tangent repeats every 180° with vertical asymptotes at 90° and 270°, and no defined amplitude. Students sketch these graphs by plotting key points, such as sin(0°)=0, sin(90°)=1, and interpret features like maximums, minimums, and zeros to solve for angles given ratios.

This topic fits within Geometry and Trigonometry, linking algebraic manipulation of ratios to visual representation. Students model real-world periodic phenomena, from Ferris wheel heights to sound waves, fostering connections between abstract functions and applications. Key skills include recognising transformations and using graphs to estimate angles, preparing for advanced calculus.

Active learning suits this topic well. When students plot points collaboratively or match graphs to scenarios, they grasp period and asymptotes through trial and error. Physical models, like swinging pendulums, make periodicity tangible, while graphing software allows experimentation with phases, turning passive sketching into dynamic discovery.

Key Questions

1. How do the graphs of sine, cosine, and tangent differ in terms of period, amplitude, and asymptotes?
2. What real-world phenomena can be modelled by periodic trigonometric graphs?
3. How can we use the graphs to find angles for a given trigonometric ratio?

Learning Objectives

• Compare the graphical representations of sine, cosine, and tangent functions, identifying differences in period, amplitude, and asymptotes.
• Sketch the graphs of y = sin(x), y = cos(x), and y = tan(x) for 0° ≤ x ≤ 360°, accurately plotting key points and indicating critical features.
• Calculate the values of trigonometric ratios for specific angles within 0° to 360° using the unit circle and graphical interpretations.
• Determine the possible values of an angle x (0° ≤ x ≤ 360°) given the value of sin(x), cos(x), or tan(x) by interpreting their respective graphs.
• Explain how periodic trigonometric functions can model real-world phenomena such as tides or alternating current.

Before You Start

Unit Circle and Trigonometric Ratios

Why: Students need to understand how sine, cosine, and tangent are defined in terms of the unit circle and their values for key angles (0°, 90°, 180°, 270°, 360°) before graphing.

Coordinate Geometry and Plotting Points

Why: The ability to plot points accurately on a Cartesian plane is fundamental to sketching any graph, including trigonometric functions.

Key Vocabulary

Period	The horizontal length of one complete cycle of a periodic function. For sine and cosine, this is 360°, and for tangent, it is 180°.
Amplitude	The measure of the maximum displacement or distance moved by a point on a vibrating body or wave measured from its equilibrium position. For sine and cosine, this is 1.
Asymptote	A line that a curve approaches but never touches. Vertical asymptotes occur for the tangent function at 90° and 270°.
Trigonometric Ratio	A ratio of the lengths of two sides in a right-angled triangle, or a ratio involving coordinates of a point on the unit circle, such as sine, cosine, and tangent.

Watch Out for These Misconceptions

Common Misconception

Sine and cosine graphs are identical.

What to Teach Instead

Sine starts at zero rising to 1 at 90°, while cosine starts at 1 falling to zero. Hands-on plotting of unit circle points reveals the 90° phase shift. Peer teaching in pairs helps students articulate and correct their graphs through comparison.

Common Misconception

Tangent has the same period as sine and cosine.

What to Teach Instead

Tangent repeats every 180°, half of sine's 360°. Graphing multiple cycles side-by-side in small groups highlights this, as students observe tan(210°)=tan(30°). Discussion clarifies why asymptotes cause the shorter repeat.

Common Misconception

All trig functions have amplitude 1.

What to Teach Instead

Sine and cosine do, but tangent approaches infinity near asymptotes. Experimenting with table values in graphing software shows unbounded growth. Collaborative prediction and plotting builds accurate mental models.

Active Learning Ideas

Point Plotting Relay: Sine and Cosine Graphs

Divide class into teams. Each student plots 5 key points for sine or cosine on graph paper, passes to next teammate for connecting and labelling max/min. Teams compare final graphs and discuss shifts between sine and cosine. Conclude with whole-class verification.

35 min·Small Groups

Tangent Asymptote Hunt: Graph Matching

Provide printed tangent graphs with hidden asymptotes. Pairs identify and mark asymptotes, periods, then sketch one full cycle. Extend by predicting tangent values near asymptotes using calculators. Share findings in a gallery walk.

30 min·Pairs

Real-World Modelling: Tide Data Graphs

Assign groups local tide height data over 24 hours. Students plot data, overlay sine model, adjust amplitude and phase to fit. Discuss how graphs predict high/low tides and solve for times using intersections.

45 min·Small Groups

Transformation Stations: Function Changes

Set up stations for amplitude, period, phase shifts on trig graphs. Groups apply one change per station using graphing tools, record before/after sketches. Rotate and explain effects to next group.

40 min·Small Groups

Real-World Connections

• Oceanographers use sine and cosine functions to model the rise and fall of tides in coastal areas, predicting high and low tide times for navigation and coastal management.
• Electrical engineers utilize cosine functions to represent alternating current (AC) voltage and current in power systems, essential for designing and maintaining the electrical grid.
• Sound engineers analyze waveforms, often represented by sine functions, to understand the properties of sound, such as frequency and amplitude, for audio production and acoustics.

Assessment Ideas

Quick Check

Provide students with a blank set of axes for 0° to 360°. Ask them to sketch the graph of y = cos(x) and label the maximum, minimum, and zero points. Then, ask them to identify the amplitude and period.

Exit Ticket

Give students a card with the equation sin(x) = 0.5. Ask them to use their knowledge of the sine graph to find two possible values for x between 0° and 360°. They should briefly explain their reasoning.

Discussion Prompt

Pose the question: 'How are the graphs of sine and tangent functions similar, and how are they different?' Guide students to discuss period, amplitude, and the presence of asymptotes, encouraging them to use precise mathematical language.

Frequently Asked Questions

How do sine, cosine, and tangent graphs differ?

Sine and cosine have 360° periods and amplitude 1; sine starts at (0,0), cosine at (0,1). Tangent has 180° period, asymptotes at odd multiples of 90°, no amplitude. Students best learn by plotting radian equivalents alongside degrees for full cycle views, noting tan's discontinuities force careful domain restrictions.

What real-world examples model trig graphs?

Ocean tides follow sine waves with daily periods; Ferris wheels model cosine heights. Sound waves and pendulum swings approximate sine. Guide students to collect data like shadow lengths for sundials, fit trig curves, and interpret parameters like amplitude for wave height predictions.

How can active learning help teach trig graphs?

Activities like relay plotting or physical models engage kinesthetic learners, making abstract periods concrete. Groups match scenarios to graphs or adjust parameters on software, revealing transformations intuitively. This builds confidence in sketching and interpreting, as peer explanations correct errors in real time during rotations.

How to use graphs to find angles for trig ratios?

Locate the y-value (ratio) on the graph, read x-intercepts or heights for angles in 0°-360°. For sine 0.5, find 30° and 150°. Practice with ambiguous case discussions; students trace horizontal lines on personal sketches to visualise multiple solutions, reinforcing principal values.