Mathematics · Class 11

Active learning ideas

Graphs of Other Trigonometric Functions

Beyond the predictable waves of sine and cosine lies a more dramatic landscape. Let's explore the graphs of the other four trigonometric functions, where we'll encounter infinite peaks and sudden breaks.

• CBSE Learning Outcomes
• NCERT Class 11: Chapter 3 - Trigonometric Functions

• 30–40 min
• Pairs → Whole Class
• 3 activities

Activity 01

• Simulation Game
• 40 min · Pairs

Graphing from the Unit Circle

Students use a large printout of the unit circle. They plot the value of tan(θ) = y/x for various angles θ, noticing how the value grows infinitely large as θ approaches π/2, thus discovering the concept of an asymptote organically.

Explain why the graph of y = tan(x) has vertical asymptotes.

Facilitation Tip

Encourage students to focus on what happens when the x-coordinate (cos θ) gets very close to zero.

What to look for

Exit Ticket: Ask students to write down the domain of y = cot(x) and the equation of one of its vertical asymptotes.

• Apply
• Analyze
• Evaluate
• Create
• Social Awareness
• Decision-Making

Activity 02

• Simulation Game
• 30 min · Individual

Reciprocal Graph Sketching

Provide students with an accurate graph of y = sin(x). On the same axes, they sketch y = cosec(x) by plotting the reciprocal of the y-values at key points (e.g., if sin(x)=1/2, cosec(x)=2).

Compare the domain and range of y = sec(x) with that of y = cos(x).

Facilitation Tip

Ask guiding questions like, 'What happens to 1/y when y gets very close to zero?' to help them discover the asymptotes.

What to look for

A section in a unit test where students must match the four functions (tan, cot, sec, cosec) to their respective graphs and list the period and range for each.

• Apply
• Analyze
• Evaluate
• Create
• Social Awareness
• Decision-Making

Activity 03

• Simulation Game
• 35 min · Small Groups

Digital Graph Explorer

Using a free graphing tool like GeoGebra or Desmos, students plot pairs like y = cos(x) and y = sec(x) simultaneously. They can then observe the relationships between the zeroes of one function and the asymptotes of the other.

Analyse the relationship between the graphs of y = sin(x) and y = cosec(x).

Facilitation Tip

Provide a worksheet with prompts like 'Change the period of cos(x). How does the graph of sec(x) change?'

What to look for

Provide a checklist with statements like 'I can explain why the range of sec(x) does not include 0.5' for students to rate their own confidence.

• Apply
• Analyze
• Evaluate
• Create
• Social Awareness
• Decision-Making

A few notes on teaching this unit

Begin by deriving the graph of tan(x) directly from sin(x) and cos(x) on the same axes. Emphasise the relationship: where cos(x) is zero, tan(x) is undefined, creating an asymptote. Then, introduce sec(x) by visually taking the reciprocal of the y-values of the cos(x) graph. This hands-on approach builds a strong conceptual link between the functions.

Your students will learn to sketch these complex graphs and precisely describe their key features, including their unique periods, domains, ranges, and the vertical asymptotes that define them.

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Watch Out for These Misconceptions

• The period of all trigonometric functions is 2π.

  The period of a function is the length of one complete cycle. For tan(x) and cot(x), the pattern of values repeats every π radians, not 2π. This is because tan(x+π) = sin(x+π)/cos(x+π) = (-sin(x))/(-cos(x)) = tan(x).

• The range of sec(x) and cosec(x) is all real numbers, just like tan(x).

  Since sec(x) = 1/cos(x) and the value of cos(x) is always between -1 and 1 (inclusive), its reciprocal, sec(x), can never be a value between -1 and 1 (exclusive). The range is (-∞, -1] U [1, ∞).

• Asymptotes are part of the graph that the function touches at infinity.

  A vertical asymptote is a line that the graph approaches but never touches or crosses. It represents an x-value for which the function is undefined, typically because it would involve division by zero.

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Methods used in this brief

• Simulation Game