1st Year Math Chapter 11 Exercise 11.2 Notes: Graphs of Sine, Cosine, and Tangent (Punjab Board 2026-27)
Complete concept notes, formulas, and solved examples for ICS/FSc Part 1 Mathematics, Punjab Board (PECTAA), Single National Curriculum 2026-27 session.
Exercise 11.2 uses the domain, range, and period facts from Exercise 11.1 to sketch and analyze the graphs of the sine, cosine, and tangent functions.
What Does This Exercise Cover?
This exercise teaches how to identify key features — intercepts, maximum and minimum points, and asymptotes — when graphing y = sin x, y = cos x, and y = tan x.
Key Concepts
1. Graph of y = sin x
The graph of y = sin x (typically drawn from −2π to 2π) is a smooth, repeating wave oscillating between −1 and 1, passing through the origin, with x-intercepts at every multiple of π.
2. Graph of y = cos x
The graph of y = cos x has the same wave shape as sine but is shifted — it starts at its maximum value of 1 when x = 0, rather than at zero.
3. Graph of y = tan x
The graph of y = tan x (typically drawn from −π to π) consists of repeating branches that rise continuously between vertical asymptotes, located wherever cos x = 0.
4. Key Features to Identify When Graphing
For any trigonometric graph, identify the x-intercepts, the maximum and minimum points (where they exist), any vertical asymptotes, and confirm the shape repeats after one full period.
Solved Examples
Example 1: State the x-intercepts of y = sin x in the interval [0, 2π].
sin x = 0 at the start, middle, and end of the interval.
Answer: x = 0, π, 2π
Example 2: State the maximum and minimum values of y = cos x and where they occur in [0, 2π].
cos x reaches its highest value at the start and end of the interval, and its lowest value at the midpoint.
Answer: Maximum = 1 at x = 0 and x = 2π; Minimum = −1 at x = π
Example 3: Identify the vertical asymptotes of y = tan x in the interval (−π, π).
tan x is undefined wherever cos x = 0.
Within (−π, π), this occurs at x = −π/2 and x = π/2.
Answer: Vertical asymptotes at x = −π/2 and x = π/2
Sample MCQs
1. The graph of y = sin x passes through the origin because:
a) sin 0 = 0 b) sin 0 = 1 c) cos 0 = 0 d) tan 0 = 1
Answer: a) sin 0 = 0
2. The maximum value of y = cos x (in [0, 2π]) occurs at:
a) x = π/2 b) x = 0 c) x = π d) x = 3π/2
Answer: b) x = 0
3. The graph of y = tan x has vertical asymptotes where:
a) sin x = 0 b) cos x = 0 c) tan x = 0 d) x = 0
Answer: b) cos x = 0
4. Between 0 and 2π, the graph of y = sin x crosses the x-axis:
a) Once b) Twice c) Three times d) Never
Answer: c) Three times
5. The graph of y = tan x repeats every:
a) π/2 b) π c) 2π d) 4π
Answer: b) π
Important Short Questions
- State the x-intercepts of y = cos x in the interval [0, 2π].
- Where does the graph of y = sin x reach its maximum value in [0, 2π]?
- Explain why the graph of y = tan x has vertical asymptotes.
- Describe the general shape of the graph of y = sin x.
- What is the value of y = tan x at x = 0?
Important Long Questions
- Sketch the graph of y = sin x from −2π to 2π, labeling key intercepts and turning points.
- Sketch the graph of y = cos x from −2π to 2π, labeling key intercepts and turning points.
- Describe the graph of y = tan x from −π to π, including its asymptotes and behavior near them.
- Compare the graphs of y = sin x and y = cos x, explaining how one relates to the other.
How to Approach This Exercise Effectively
- Plot the same handful of key x-values (0, π/2, π, 3π/2, 2π) for both sine and cosine — memorizing their values there makes sketching both graphs fast.
- Remember cosine is really sine shifted left by π/2 — if you know one graph well, the other follows immediately.
- For tangent, mark the asymptotes first, then sketch the rising branch between each pair — trying to plot points before locating asymptotes often leads to a distorted graph.
- Always double-check your intercepts and turning points against the domain/range facts from Exercise 11.1 before finalizing a sketch.
- Practice sketching all three graphs from memory repeatedly — recognizing their shape instantly saves significant time in exams.
FAQs
Q: What is the difference between the graphs of sine and cosine?
A: They have identical wave shapes, but cosine’s graph is shifted horizontally compared to sine’s — cosine starts at its maximum while sine starts at zero.
Q: Why does the tangent graph have asymptotes but sine and cosine don’t?
A: Because tangent is undefined wherever cosine equals zero, creating vertical lines the graph approaches but never crosses; sine and cosine have no such undefined points.
Q: How many times does y = sin x cross the x-axis in one full period?
A: It crosses three times within a closed interval of one period (for example, at 0, π, and 2π), though only two of these mark the start and end of a single repeating cycle.
Q: What real-world phenomena are modeled by sine and cosine graphs?
A: Examples include sound waves, alternating electrical current, tides, and any smoothly repeating oscillation over time.
Notes prepared for Punjab Board (PECTAA) 1st Year Mathematics, 2026-27 session, Chapter 11: Trigonometric Functions and their Graphs, Exercise 11.2.