Mathematics · Class 11 · Trigonometric Functions · Term 3

Graphs of Other Trigonometric Functions

Analyse the graphs of tangent, cotangent, secant, and cosecant functions, paying close attention to their domain, range, period, and vertical asymptotes.

TL;DR: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

About This Topic

This topic extends students' understanding of trigonometric functions beyond the familiar sine and cosine waves, aligning with the CBSE and other state board curricula for Class 11. The focus shifts to functions derived from ratios and reciprocals: tangent, cotangent, secant, and cosecant. The core pedagogical challenge is moving from the continuous, bounded nature of sine and cosine to functions with discontinuities, specifically vertical asymptotes. This introduces students to a new and crucial feature of graphs.

By analysing these functions, students deepen their understanding of fundamental concepts like domain, range, and periodicity. For instance, they discover that the period of tangent and cotangent is π, not 2π, a key differentiator. This topic is foundational for Class 12 calculus, where the derivatives and integrals of these functions are studied, and their asymptotic behaviour is critical for understanding limits. It also reinforces the relationships between the six trigonometric functions, moving them from abstract identities to visual, graphical connections.

Key Questions

1. Explain why the graph of y = tan(x) has vertical asymptotes.
2. Compare the domain and range of y = sec(x) with that of y = cos(x).
3. Analyse the relationship between the graphs of y = sin(x) and y = cosec(x).

Learning Objectives

• Sketch the graphs of tangent, cotangent, secant, and cosecant functions.
• Identify the domain, range, period, and equations of vertical asymptotes for each function.
• Analyse the relationship between the graphs of a function and its reciprocal, for example, cosine and secant.
• Explain how the zeroes of sine and cosine functions relate to the vertical asymptotes of their reciprocal and quotient functions.
• Solve simple trigonometric equations graphically using the properties of these functions.

Key Vocabulary

Vertical Asymptote	A vertical line x = a that the graph of a function approaches but never touches, occurring where the function is undefined.
Period	The smallest positive value 'p' for which f(x+p) = f(x) for all x. It is the horizontal length of one complete cycle of the graph.
Domain	The set of all possible input values (x-values) for which a function is defined.
Range	The set of all possible output values (y-values) that a function can produce.
Reciprocal	The multiplicative inverse of a number or function. For example, the reciprocal of sin(x) is 1/sin(x), which is cosec(x).

Watch Out for These Misconceptions

Common Misconception

The period of all trigonometric functions is 2π.

What to Teach Instead

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).

Common Misconception

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

What to Teach Instead

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, ∞).

Common Misconception

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

What to Teach Instead

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.

Active Learning Ideas

Simulation Game

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.

40 min·Pairs

Simulation Game

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).

30 min·Individual

Simulation Game

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.

35 min·Small Groups

Real-World Connections

• In electronics, the impedance of certain AC circuits can exhibit behaviour similar to the tangent function, approaching infinity at resonant frequencies.
• In physics and optics, Brewster's angle, which relates to light polarization, is calculated using the arctangent function, the inverse of the tangent.
• Some oscillating systems in mechanical engineering that have periodic 'blow-ups' or infinite responses can be modelled using secant or cosecant functions.
• In architecture and design, patterns that repeat but have regular breaks or interruptions can be conceptualised using these discontinuous trigonometric functions.

Assessment Ideas

Exit Ticket

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

Quick Check

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.

Quick Check

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.

Frequently Asked Questions

Why does the graph of y = tan(x) have vertical asymptotes?

The function tan(x) is defined as sin(x)/cos(x). Division by zero is undefined. The graph has vertical asymptotes at the x-values where the denominator, cos(x), is equal to zero. This occurs at x = π/2, 3π/2, 5π/2, and so on, or more generally at x = (2n+1)π/2 where n is any integer.

What is the main difference between the domain of y = sec(x) and y = cos(x)?

The domain of y = cos(x) is all real numbers, as it is a continuous wave. However, y = sec(x) is 1/cos(x). Its domain excludes all values of x for which cos(x) = 0. Therefore, the domain of sec(x) is all real numbers except odd multiples of π/2.

How are the graphs of y = sin(x) and y = cosec(x) related?

They are reciprocal graphs. The y-values of cosec(x) are the reciprocals of the y-values of sin(x). Where sin(x) = 1, cosec(x) = 1. Where sin(x) = 0, cosec(x) is undefined, resulting in a vertical asymptote. When sin(x) is positive, cosec(x) is also positive, and vice versa.