Graphs of Trigonometric Functions

Active learning works for graphs of trigonometric functions because students must physically or digitally manipulate graphs to see patterns. Plotting points, matching shapes, and moving between representations helps students build mental models of periodicity and asymptotes that static notes cannot provide.

• Year 10
• Mathematics
• 4 activities
• 20 min–45 min

Learning Objectives

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

Gallery Walk

⏱ 30 min·Pairs

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.

Prepare & details

Compare the key features of the sine, cosine, and tangent graphs.

Facilitation Tip: During Graph Matching: Trig Functions, circulate and ask pairs to justify why they matched a specific graph, reinforcing precise vocabulary like amplitude and period.

Setup: Wall space or tables arranged around room perimeter

Materials: Large paper/poster boards, Markers, Sticky notes for feedback

• Understand
• Apply
• Analyze
• Create
• Relationship Skills
• Social Awareness

Gallery Walk

⏱ 45 min·Small Groups

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.

Prepare & details

Predict the values of sin(x), cos(x), and tan(x) for angles beyond 90 degrees using their graphs.

Facilitation Tip: For Desmos Exploration: Phase Shifts, pause the class at key moments to discuss how horizontal shifts affect the starting point of the graph.

Setup: Wall space or tables arranged around room perimeter

Materials: Large paper/poster boards, Markers, Sticky notes for feedback

• Understand
• Apply
• Analyze
• Create
• Relationship Skills
• Social Awareness

Gallery Walk

⏱ 20 min·Whole Class

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.

Prepare & details

Explain the periodic nature of trigonometric functions.

Facilitation Tip: When running Human Graph: Sine Wave, remind students that their bodies represent points on the graph, so spacing must be consistent with the function's period.

Setup: Wall space or tables arranged around room perimeter

Materials: Large paper/poster boards, Markers, Sticky notes for feedback

• Understand
• Apply
• Analyze
• Create
• Relationship Skills
• Social Awareness

Gallery Walk

⏱ 25 min·Pairs

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.

Prepare & details

Compare the key features of the sine, cosine, and tangent graphs.

Facilitation Tip: Use Key Features Sort: Comparison Cards to prompt students to verbalize why sine and cosine are not identical before shifting into the tangent card sort.

Setup: Wall space or tables arranged around room perimeter

Materials: Large paper/poster boards, Markers, Sticky notes for feedback

• Understand
• Apply
• Analyze
• Create
• Relationship Skills
• Social Awareness

Teaching This Topic

Experienced teachers approach this topic by connecting graphs to the unit circle first, then using technology to animate transformations. Avoid rushing to formulas; instead, build intuition through multiple representations. Research suggests students grasp periodicity better when they observe how sine and cosine waves repeat every 360 degrees, while tangent's 180-degree cycle emerges from its definition in terms of sine and cosine.

What to Expect

Successful learning looks like students confidently sketch sine, cosine, and tangent graphs, accurately labeling amplitude, period, and key points. They should explain differences between the functions and predict values using the unit circle without relying on memorized rules alone.

Watch Out for These Misconceptions

Common Misconception

During Graph Matching: Trig Functions, watch for students who claim sine and cosine graphs are identical but shifted.

What to Teach Instead

Hand pairs two unlabeled graphs and ask them to compare starting points and zeros. Have them mark where each graph crosses the x-axis and identify the y-value at x=0 to highlight differences before matching.

Common Misconception

During Desmos Exploration: Phase Shifts, watch for students who assume tangent has the same period as sine and cosine.

What to Teach Instead

Use the graphing tool to zoom in on one period of tangent and ask students to count repetitions from 0 to 360 degrees. Guide them to observe that tangent repeats twice in this interval while sine and cosine repeat once.

Common Misconception

During Human Graph: Sine Wave, watch for students who assume all trigonometric functions are defined for all angles.

What to Teach Instead

After forming the sine wave, pause at odd multiples of 90 degrees and ask students to consider the value of tangent at those points. Direct their attention to the asymptotes in the graph cards to connect the physical representation to the discontinuities.

Assessment Ideas

Quick Check

After Graph Matching: Trig Functions, 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

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

Discussion Prompt

During Key Features Sort: Comparison Cards, 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.

Extensions & Scaffolding

• Challenge students to predict and sketch y = 2sin(x) and y = sin(2x) after completing the Desmos Exploration, explaining how each transformation affects the graph.
• For students who struggle, provide pre-labeled graphs with blanks for amplitude, period, and key points to complete during Graph Matching: Trig Functions.
• Deeper exploration: Ask students to research and present how trigonometric functions model real-world phenomena like sound waves or tides, linking phase shifts to practical applications.

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).