Graphs of Trigonometric Functions

Active learning breaks down the abstract nature of trigonometric graphs by letting students physically plot points, compare shapes, and manipulate functions. When students move from calculating ratios to drawing curves, they connect unit circle values to visual patterns, making periodicity and amplitude concrete. Movement and collaboration also help students correct their own misconceptions through immediate peer feedback.

• Secondary 4
• Mathematics
• 4 activities
• 30 min–45 min

Learning Objectives

1. Compare the graphical representations of sine, cosine, and tangent functions, identifying differences in period, amplitude, and asymptotes.
2. 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.
3. Calculate the values of trigonometric ratios for specific angles within 0° to 360° using the unit circle and graphical interpretations.
4. 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.
5. Explain how periodic trigonometric functions can model real-world phenomena such as tides or alternating current.

⏱ 35 min·Small Groups

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.

Prepare & details

How do the graphs of sine, cosine, and tangent differ in terms of period, amplitude, and asymptotes?

Facilitation Tip: During the Point Plotting Relay, rotate students between stations every 3 minutes so they plot points collaboratively, building the sine curve quadrant by quadrant.

⏱ 30 min·Pairs

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.

Prepare & details

What real-world phenomena can be modelled by periodic trigonometric graphs?

Facilitation Tip: For Tangent Asymptote Hunt, provide tracing paper so students can overlay graphs and visually confirm the 180° period and asymptote locations.

⏱ 45 min·Small Groups

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.

Prepare & details

How can we use the graphs to find angles for a given trigonometric ratio?

Facilitation Tip: In Transformation Stations, place a completed parent graph at each station so students compare changes directly instead of recreating base graphs from memory.

⏱ 40 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.

Prepare & details

How do the graphs of sine, cosine, and tangent differ in terms of period, amplitude, and asymptotes?

Facilitation Tip: During Real-World Modelling, provide graph paper with pre-labeled tide times so students focus on scaling and interpolation rather than axis setup.

Teaching This Topic

Start with quick sketches on whiteboards to surface prior knowledge and misconceptions before formal instruction. Use color coding—blue for sine, red for cosine—to help students visually separate the two graphs. Avoid teaching phase shifts until students can confidently graph the parent functions, as this prevents confusion about starting points. Research shows that tactile plotting improves retention more than calculator demonstrations, so prioritize hands-on time over screen time.

What to Expect

Students will confidently sketch sine, cosine, and tangent graphs from 0° to 360°, identify key features like maxima, minima, zeros, and asymptotes, and explain how transformations change these features. They will use correct mathematical language to compare periods, amplitudes, and behaviors of each function. Peer discussions and quick checks ensure accuracy before moving forward.

Watch Out for These Misconceptions

Common Misconception

During Point Plotting Relay, watch for students who assume sine and cosine graphs are identical because they both oscillate between -1 and 1.

What to Teach Instead

Have pairs compare their plotted points side-by-side and label each axis with the corresponding unit circle angle. Ask them to point out where sine starts at 0 and cosine starts at 1, using the unit circle as a reference to reinforce the phase shift.

Common Misconception

During Tangent Asymptote Hunt, watch for students who assume tangent has the same period as sine and cosine.

What to Teach Instead

Ask students to graph two full cycles of tangent side-by-side and mark tan(30°) and tan(210°). Have them observe that the values repeat halfway through the cycle, then discuss how asymptotes cause this shorter period.

Common Misconception

During Transformation Stations, watch for students who generalize that all trigonometric functions have an amplitude of 1.

What to Teach Instead

Direct students to the tangent station and ask them to graph y = 2tan(x) and y = 0.5tan(x). Have them observe that the amplitude concept does not apply to tangent due to its asymptotes, then sketch the unbounded behavior near 90° and 270°.

Assessment Ideas

Quick Check

After Point Plotting Relay, 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

After Tangent Asymptote Hunt, 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

During Real-World Modelling, 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.

Extensions & Scaffolding

• Challenge advanced students to predict how changing the amplitude or period affects tide graphs by altering the equation and re-plotting.
• Scaffolding for struggling students: Provide a completed sine graph with blanks for key points; ask them to fill in the missing values and explain why each point belongs there.
• Deeper exploration: Ask students to research and compare the graphs of cotangent, secant, and cosecant, noting similarities to tangent, cosine, and sine respectively.

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.