Mathematics · Year 10 · Advanced Geometry and Measures · Summer Term

Graphs of Trigonometric Functions

Sketching and interpreting graphs of y = sin(x), y = cos(x), and y = tan(x).

• National Curriculum Attainment Targets
• GCSE: Mathematics - Geometry and Measures

About This Topic

Graphs of trigonometric functions introduce students to sketching and interpreting y = sin(x), y = cos(x), and y = tan(x), key elements of the GCSE Geometry and Measures strand. Students identify amplitude, period, and key points: sine and cosine oscillate between -1 and 1 with a 360-degree period, sine starting at (0,0) and cosine at (0,1), while tangent has a 180-degree period with vertical asymptotes at odd multiples of 90 degrees. These graphs extend understanding of trigonometry from right triangles to the unit circle, enabling predictions of function values beyond 90 degrees and explanations of periodicity.

This topic connects graphing skills to real-world applications like waves, tides, and oscillations, reinforcing fluency in angles and measures. Comparing features across functions builds analytical skills essential for solving trigonometric equations and modelling periodic phenomena later in the curriculum.

Active learning suits this topic well. Students manipulate graphs through interactive tools or physical plotting, making abstract periodicity visible and fostering deeper retention through peer discussion and prediction tasks.

Key Questions

1. Compare the key features of the sine, cosine, and tangent graphs.
2. Predict the values of sin(x), cos(x), and tan(x) for angles beyond 90 degrees using their graphs.
3. Explain the periodic nature of trigonometric functions.

Learning Objectives

• Compare the amplitude, period, and key intercepts of the sine, cosine, and tangent graphs.
• Predict the value of sin(x), cos(x), and tan(x) for angles between 0 and 360 degrees using their respective graphs.
• Explain the concept of periodicity for sine, cosine, and tangent functions by identifying repeating patterns in their graphs.
• Identify the location and behavior of vertical asymptotes on the tangent graph.

Before You Start

Coordinates and Graphing Straight Lines

Why: Students need to be comfortable plotting points and understanding the Cartesian coordinate system to sketch and interpret function graphs.

Angles and Degrees

Why: A solid understanding of angle measurement in degrees, including values beyond 90 degrees, is essential for working with trigonometric functions.

Basic Trigonometry (SOH CAH TOA)

Why: Familiarity with sine, cosine, and tangent ratios in right-angled triangles provides a foundation for understanding these functions in a broader context.

Key Vocabulary

Amplitude	For sine and cosine graphs, this is half the distance between the maximum and minimum values of the function. It represents the maximum displacement from the horizontal axis.
Period	The horizontal length of one complete cycle of a periodic function. For sine and cosine, this is 360 degrees; for tangent, it is 180 degrees.
Vertical Asymptote	A vertical line that the graph of a function approaches but never touches. These occur at specific x-values for the tangent function.
Intercept	The points where a graph crosses the x-axis (x-intercepts) or the y-axis (y-intercept).

Watch Out for These Misconceptions

Common Misconception

Sine and cosine graphs are identical, just shifted.

What to Teach Instead

Sine starts at (0,0) with zeros at multiples of 180 degrees, while cosine peaks at (0,1) with zeros offset by 90 degrees. Graph-matching activities in pairs help students visually compare and articulate differences, building accurate mental models.

Common Misconception

Tangent graph has the same period as sine and cosine.

What to Teach Instead

Tangent repeats every 180 degrees due to its definition, unlike the 360-degree period of sine and cosine. Exploration with graphing tools allows students to observe repetitions and asymptotes firsthand, clarifying through guided peer questioning.

Common Misconception

Trigonometric functions are defined for all angles.

What to Teach Instead

Tangent is undefined at odd multiples of 90 degrees, shown by asymptotes. Physical plotting or software demos make these discontinuities tangible, prompting discussions that correct overgeneralisation from sine and cosine.

Active Learning Ideas

Graph Matching: Trig Functions

Prepare cards with equations, graphs, and descriptions of sin, cos, tan. In pairs, students match sets and justify choices by noting amplitude, period, and key points. Follow with a class share-out to verify matches.

30 min·Pairs

Desmos Exploration: Phase Shifts

Using Desmos, small groups input y=sin(x), y=cos(x), y=tan(x) and adjust sliders for amplitude and period. They sketch predictions first, then compare outputs and note changes in key features.

45 min·Small Groups

Human Graph: Sine Wave

Mark axes on the floor with tape. Whole class students hold cards with x,y values for y=sin(x) from 0 to 360 degrees, forming the graph. Discuss shape, then repeat for cosine.

20 min·Whole Class

Key Features Sort: Comparison Cards

Distribute cards listing features like 'period 180 degrees' or 'asymptotes at 90 degrees'. Individuals or pairs sort into sin, cos, tan columns, then explain reasoning to the group.

25 min·Pairs

Real-World Connections

• Electrical engineers use sine and cosine waves to represent alternating current (AC) electricity, modeling voltage and current fluctuations over time.
• Oceanographers use trigonometric functions to model tidal patterns, predicting the rise and fall of sea levels at coastal locations like the Bay of Fundy.

Assessment Ideas

Quick Check

Provide students with printed graphs of y = sin(x), y = cos(x), and y = tan(x) for x from 0 to 360 degrees. Ask them to label the period of each graph and identify one x-intercept for each.

Exit Ticket

On an exit ticket, ask students to sketch a rough graph of y = cos(x) for 0 to 360 degrees. Then, ask them to state the value of cos(90) and cos(270) based on their sketch and explain why the tangent graph has vertical asymptotes.

Discussion Prompt

Pose the question: 'How are the sine and cosine graphs similar, and how are they different?' Encourage students to discuss features like starting points, amplitude, and period. Then, ask how the tangent graph fundamentally differs from the other two.

Frequently Asked Questions

What are the key features of sine, cosine, and tangent graphs for Year 10?

Sine and cosine have amplitude 1, period 360 degrees; sine passes through origin, cosine through (0,1). Tangent has period 180 degrees, range all reals, asymptotes at 90 + 180k degrees. Teaching focuses on sketching from 0-360 for sine/cosine, -90 to 90 for tangent, using tables of values to plot accurately.

How to teach sketching trig graphs GCSE Maths?

Start with unit circle values for key angles (0,30,45,60,90 etc.), plot points, connect smoothly. Emphasise symmetry and transformations. Practice progresses from blank axes to interpreting scales, with peer review ensuring precision in periods and amplitudes.

How can active learning help students understand trig graphs?

Activities like human graphs or Desmos sliders engage kinesthetic and visual learners, turning static sketches into dynamic explorations. Pairs predicting then verifying features discuss misconceptions in real time, while group rotations build collaborative graphing fluency essential for GCSE exam questions on interpretation.

Common misconceptions in trig function graphs Year 10?

Students often mix sine/cosine phases or overlook tangent's shorter period and asymptotes. Address via sorting tasks or matching games that require verbal justification, helping revise mental models. Regular low-stakes quizzes with graph predictions reinforce corrections over time.