Graphing Other Trigonometric Functions

Active learning works for graphing trigonometric functions because these concepts rely on visual patterns and transformations that students discover through doing. Moving beyond static images helps students internalize the relationships between equations, graphs, and key features like asymptotes and periods.

• Grade 12
• Mathematics
• 4 activities
• 20 min–45 min

Learning Objectives

1. Compare and contrast the graphical features, including period and asymptotes, of tangent and cotangent functions with those of sine and cosine functions.
2. Explain the reciprocal relationship between the graphs of secant and cosecant functions and their corresponding cosine and sine functions, identifying points of intersection and asymptotes.
3. Construct the graph of a tangent function, y = a tan(b(x - c)) + d, given its period, phase shift, vertical stretch, and vertical translation.
4. Identify the domain and range for tangent, cotangent, secant, and cosecant functions based on their graphical characteristics and asymptotes.

Stations Rotation

⏱ 30 min·Pairs

Pairs Relay: Reciprocal Graphs

Partners graph sine or cosine on shared axes, then add reciprocal secant or cosecant by plotting 1/y values. Note asymptotes where denominator nears zero. Switch roles to graph cosine/tangent pair and compare.

Prepare & details

Differentiate the graphical features and asymptotes of tangent and cotangent functions from sine and cosine.

Facilitation Tip: During Pairs Relay: Reciprocal Graphs, circulate to ensure pairs are correctly labeling asymptotes on both y=cos(x) and y=sec(x) before moving to y=sin(x) and y=csc(x).

Setup: Tables/desks arranged in 4-6 distinct stations around room

Materials: Station instruction cards, Different materials per station, Rotation timer

• Remember
• Understand
• Apply
• Analyze
• Self-Management
• Relationship Skills

Stations Rotation

⏱ 45 min·Small Groups

Small Groups: Tangent Builder

Groups receive cards with period, phase shift, and vertical stretch for tangent. They sketch graphs on mini-whiteboards, mark asymptotes every π/period, and test with Desmos. Rotate cards and critique peers' work.

Prepare & details

Explain the relationship between the graphs of secant/cosecant and their reciprocal functions.

Facilitation Tip: For Tangent Builder, provide graph paper with pre-marked axes and have students label each period clearly before adding phase shifts.

Setup: Tables/desks arranged in 4-6 distinct stations around room

Materials: Station instruction cards, Different materials per station, Rotation timer

• Remember
• Understand
• Apply
• Analyze
• Self-Management
• Relationship Skills

Stations Rotation

⏱ 25 min·Whole Class

Whole Class: Asymptote Prediction

Display base tan graph on projector. Students predict asymptote shifts for phase-changed versions, vote via thumbs up/down. Reveal animation, discuss why shifts align with period.

Prepare & details

Construct the graph of a tangent function given its period and phase shift.

Facilitation Tip: In Asymptote Prediction, ask students to write their predicted asymptote equations on mini whiteboards before revealing the actual graphs for comparison.

Setup: Tables/desks arranged in 4-6 distinct stations around room

Materials: Station instruction cards, Different materials per station, Rotation timer

• Remember
• Understand
• Apply
• Analyze
• Self-Management
• Relationship Skills

Stations Rotation

⏱ 20 min·Individual

Individual: Cotangent Match-Up

Students match cotangent equations to graphs focusing on asymptote positions and phase. Self-check with key, then pair to explain one mismatch.

Prepare & details

Differentiate the graphical features and asymptotes of tangent and cotangent functions from sine and cosine.

Facilitation Tip: During Cotangent Match-Up, ask students to explain their matching choices to their partner, focusing on how the asymptotes relate to the parent function.

Setup: Tables/desks arranged in 4-6 distinct stations around room

Materials: Station instruction cards, Different materials per station, Rotation timer

• Remember
• Understand
• Apply
• Analyze
• Self-Management
• Relationship Skills

Teaching This Topic

Teach these functions by connecting them to sine and cosine through their reciprocal definitions, emphasizing the symmetry that leads to different periods and asymptote locations. Avoid starting with transformations; instead, build from the parent functions so students see why the features emerge. Use technology like Desmos to animate the reciprocal relationship, but always require students to sketch by hand to solidify understanding.

What to Expect

Successful learning looks like students accurately identifying periods, asymptotes, and transformations without relying on rote memorization. They should discuss patterns with peers, justify their reasoning with sketches or equations, and apply their understanding to new problems with confidence.

Watch Out for These Misconceptions

Common Misconception

During Pairs Relay: Reciprocal Graphs, watch for students who assume tangent and cotangent have the same period as sine and cosine.

What to Teach Instead

Prompt pairs to count the number of full cycles in a 2π interval on their graphs and compare this to the sine and cosine graphs they’ve already seen.

Common Misconception

During Pairs Relay: Reciprocal Graphs, watch for students who think secant is a reflection of cosine over the x-axis.

What to Teach Instead

Have students overlay the two graphs in Desmos and trace the behavior of y=1/cos(x) near the zeros of cosine to see the asymptotes and hyperbolas form.

Common Misconception

During Cotangent Match-Up, watch for students who place cotangent asymptotes in the same locations as tangent.

What to Teach Instead

Ask students to write the equation of the parent function for each graph they match and explain how the asymptotes shift due to the reciprocal relationship.

Assessment Ideas

Exit Ticket

After Tangent Builder, collect students’ sketches for y = 3 tan(2x - π). Assess their ability to identify the period (π/2), phase shift (π/2 right), and two consecutive asymptotes (x = π/2 and x = π).

Quick Check

During Asymptote Prediction, display y = sec(x) and y = csc(x) graphs side-by-side. Ask students to write the parent function (cosine for secant, sine for cosecant) and explain in one sentence why asymptotes appear where cos(x)=0 or sin(x)=0.

Discussion Prompt

After Cotangent Match-Up, pose the question: 'How does the absence of a maximum or minimum value in cotangent functions affect its graph compared to sine and cosine?' Facilitate a class discussion where students share their reasoning and examples from the activity.

Extensions & Scaffolding

• Challenge: Ask students to derive the period of y = tan(3x - π/4) and sketch two full periods without graphing software.
• Scaffolding: Provide students with a partially completed cotangent graph and ask them to fill in the asymptotes and key points using the parent function.
• Deeper exploration: Have students research real-world applications of secant or cosecant functions (e.g., in physics or engineering) and present their findings to the class.

Key Vocabulary

Vertical Asymptote	A vertical line that the graph of a function approaches but never touches. For tangent and cotangent, these occur at multiples of pi/2 and pi respectively, where the denominator of the reciprocal function is zero.
Period	The horizontal length of one complete cycle of a periodic function. The period of tangent and cotangent is pi, while the period of secant and cosecant is 2pi.
Reciprocal Functions	Pairs of functions where one is the reciprocal of the other, such as secant and cosine (sec x = 1/cos x) or cosecant and sine (csc x = 1/sin x). Their graphs are related through asymptotes and points of intersection.
Phase Shift	The horizontal displacement of a periodic function from its parent graph. For tangent functions, this is represented by the 'c' value in y = a tan(b(x - c)) + d.