To sketch the trigonometry graphs of the functions – Sine, Cosine and Tangent, we need to know the period, phase, amplitude, maximum and minimum turning points. These graphs are used in many areas of engineering and science. Few of the examples are the growth of animals and plants, engines and waves, etc. Also, we have graphs for all the trigonometric functions.
The graphical representation of sine, cosine and tangent functions are explained here briefly with the help of the corresponding graph. Students can learn how to graph a trigonometric function here along with practice questions based on it.
Table of Contents
Graphs of Trigonometric Functions
Sine, Cosine and tangent are the three important trigonometry ratios, based on which functions are defined. Below are the graphs of the three trigonometry functions sin x, cos x, and tan x. In these trigonometry graphs, x-axis values of the angles are in radians, and on the y-axis, its f(x) is taken, the value of the function at each given angle.
Sin Graph
- y = sin x
- The roots or zeros of y = sin x is at the multiples of π
- The sin graph passes the x-axis as sin x = 0 there
- Period of the sine function is 2π
- The height of the curve at each point is equal to the line value of sine
| Max value of Graph | Min value of the graph |
| 1 at π/2 | -1 at (3π/2) |
Cos Graph
- y = cos x
- sin (x + π/2 ) = cos x
- y = cos x graph is the graph we get after shifting y = sin x to π/2 units to the left
- Period of the cosine function is 2π
| Max value of Graph | Min value of the graph |
| 1 at 0, 4π | -1 at 2π |
There are a few similarities between the sine and cosine graphs, They are:
- Both have the same curve which is shifted along the x-axis
- Both have an amplitude of 1
- Have a period of 360° or 2π radians
The combined graph of sine and cosine function can be represented as follows.
Tan Graph
The tan function is completely different from sin and cos function. The function here goes between negative and positive infinity, crossing through 0 over a period of π radian.
- y = tan x
- The tangent graph has an undefined amplitude as the curve tends to infinity
- It also has a period of 180°, i.e. π
Graphs of Trigonometric Functions
The six trigonometric functions are:
- Sine
- Cosine
- Tangent
- Cosecant
- Secant
- Cotangent
Trigonometric graphs for these Trigonometry functions can be drawn if you know the following:
Amplitude
- It is the absolute value of any number multiplied with it on the trigonometric function.
- The height from the centre line to the peak (or trough) is called amplitude.
- You can also measure the height from highest to lowest points and then dividing it by 2.
- It basically tells how tall or short the curve is.
- Also, notice that the function is in usual orientation or upside down depending on the minus or plus sign of the amplitude value.
Period
The Period goes from any point (one peak) to the next matching point.
The graphical representation of period and amplitude of a function is given below.
Phase
How far the function is shifted from the usual position horizontally is called a Phase.
- Max and min turning points.
The above terms are also important to use the graph of trigonometry formulas.
How to Draw the Graph of a Trigonometric Function?
Different methods can be used to draw the graph of a trigonometric function. The detailed explanation of one of the efficient methods is given below.
While drawing a graph of the sine function, convert the given function to the general form as a sin (bx – c) + d in order to find the different parameters such as amplitude, phase shift, vertical shift and period.
Where,
|a| = Amplitude
2π/|b| = Period
c/b = Phase shift
d = Vertical shift
Similarly, for the cosine function we can use the formula a cos (bx – c) + d.
Thus, the graphs of all the six trigonometric functions are as shown in the below figure.
Graphing Trigonometric Functions Practice
Let’s practice what we learned in the above paragraphs with few of trigonometry functions graphing questions.
1) Sketch the graph of y = 5 sin 2x° + 4
- Amplitude = 5, so the distance between the max and min value is 10.
- Number of waves = 2 (Each wave has a period of 360° ÷ 2 = 180°)
- moved up by 4
- max turning point when (5 × 1)+ 4 = 9 and min turning point when (5 × -1) + 4 = -1
- Period = 2π/2 = π
- The graph looks like:
2) Sketch the graph of y = 4 cos 3x° + 7
- Amplitude = 4, so the distance between the max and min value is 8.
- Number of waves = 2 (Each wave has a period of 360° ÷ 2 = 180°)
- the vertical shift is 7
- max turning point when (4 × 1)+ 7 = 11 and min turning point when (4 × -1) + 7 = 3
- Period = 2π/3
- The graph looks like:
MCQs for Trigonometry Graphs for Sine, Cosine & Tangent Functions
Here are some Multiple Choice Questions (MCQs) on Trigonometry Graphs for Sine, Cosine & Tangent Functions for Class 11 Math & JEE Preparation:
1. What is the amplitude of the sine function y = sin(x)?
A) 0
B) 1
C) -1
D) 2
✅ Answer: B) 1
2. What is the period of the cosine function y = cos(x)?
A) π
B) 2π
C) π/2
D) 4π
✅ Answer: B) 2π
3. At what x-values does the sine function y = sin(x) intersect the x-axis?
A) x = π/2, 3π/2, 5π/2
B) x = π, 2π, 3π
C) x = 0, π, 2π, 3π
D) x = 1, 2, 3
✅ Answer: C) x = 0, π, 2π, 3π
4. What is the range of the function y = sin(x)?
A) (-∞, ∞)
B) [-1, 1]
C) [0, 1]
D) (-1, 1)
✅ Answer: B) [-1, 1]
5. What happens to the graph of y = sin(x) when the function is transformed to y = 2sin(x)?
A) The amplitude doubles
B) The period doubles
C) The function shifts left
D) The function shifts right
✅ Answer: A) The amplitude doubles
6. What is the period of the tangent function y = tan(x)?
A) 2π
B) π
C) π/2
D) 4π
✅ Answer: B) π
7. At which points does the tangent function y = tan(x) have vertical asymptotes?
A) x = 0, π, 2π, 3π
B) x = π/2, 3π/2, 5π/2
C) x = 1, 2, 3
D) x = 0, 1, 2
✅ Answer: B) x = π/2, 3π/2, 5π/2
8. How does the graph of y = cos(x) differ from the graph of y = sin(x)?
A) It is shifted π/2 units to the right
B) It is shifted π/2 units to the left
C) It has a different amplitude
D) It has a different period
✅ Answer: A) It is shifted π/2 units to the right
Topic Wise MCQs of Trigonometry Graphs
Here are some multiple-choice questions (MCQs) related to the graphs of sine, cosine, and tangent functions in trigonometry:
Sine Function (y = sin x)
- What is the amplitude of the graph of y = sin x?
a) 1
b) 2
c) π
d) 0- Answer: a) 1
- What is the period of the graph of y = sin x?
a) π
b) 2π
c) 1
d) 0- Answer: b) 2π
- At what point does the graph of y = sin x start?
a) (0, 1)
b) (0, 0)
c) (π/2, 1)
d) (π, 0)- Answer: b) (0, 0)
Cosine Function (y = cos x)
- What is the amplitude of the graph of y = cos x?
a) 1
b) 2
c) π
d) 0- Answer: a) 1
- What is the period of the graph of y = cos x?
a) π
b) 2π
c) 1
d) 0- Answer: b) 2π
- At what point does the graph of y = cos x start?
a) (0, 1)
b) (0, 0)
c) (π/2, 1)
d) (π, 0)- Answer: a) (0, 1)
Tangent Function (y = tan x)
- What is the period of the graph of y = tan x?
a) π
b) 2π
c) 1
d) 0- Answer: a) π
- What are the asymptotes of the graph of y = tan x?
a) x = π/2 + nπ, where n is an integer
b) x = nπ, where n is an integer
c) y = π/2 + nπ, where n is an integer
d) y = nπ, where n is an integer- Answer: a) x = π/2 + nπ, where n is an integer
- What is the range of the graph of y = tan x?
a) (-∞, ∞)
b) [-1, 1]
c) [0, ∞)
d) (-∞, 0]- Answer: a) (-∞, ∞)
General Questions
- Which of the following functions has a graph that passes through the origin (0, 0)?
a) y = sin x
b) y = cos x
c) y = tan x
d) Both a and c- Answer: d) Both a and c
- Which of the following functions has a graph with a maximum value of 1 and a minimum value of -1?
a) y = sin x
b) y = cos x
c) y = tan x
d) Both a and b- Answer: d) Both a and b
- Which of the following functions has a graph that is symmetric about the origin?
a) y = sin x
b) y = cos x
c) y = tan x
d) Both a and c- Answer: d) Both a and c
- What is the phase shift of the graph of y = sin(x – π/2)?
a) π/2 to the right
b) π/2 to the left
c) π to the right
d) π to the left- Answer: a) π/2 to the right
- What is the vertical shift of the graph of y = cos x + 3?
a) 3 units up
b) 3 units down
c) 3 units to the right
d) 3 units to the left- Answer: a) 3 units up
- Which of the following functions has a graph that repeats every π units?
a) y = sin x
b) y = cos x
c) y = tan x
d) Both a and b- Answer: c) y = tan x
Case Study: Understanding Trigonometry Graphs – Sine, Cosine & Tangent Functions
Ravi, a Class 11 student preparing for JEE, is learning about trigonometric function graphs. His teacher asks him to analyze the behavior of sine, cosine, and tangent functions by observing their amplitude, period, symmetry, and asymptotes. To visualize the concepts, Ravi is given the following equations:
- y=sin(x)
- y=cos(x)
- y=tan(x)
Using graphing software, he plots these functions over the interval x∈[−2π,2π] and observes their properties.
Observations & Questions:
1. Amplitude & Range Analysis
Ravi notices that sine and cosine functions oscillate between -1 and 1, while the tangent function extends infinitely.
🔹 Question 1: What is the range of the tangent function y=tan(x) ?
A) [−1,1]
B) (−∞,∞)
C) [0,∞)
D) (−1,1]
✅ Answer: B) (−∞,∞)
2. Periodicity of Trigonometric Functions
Ravi observes that the sine and cosine graphs repeat every 2π2\pi2π, while the tangent function repeats every π\piπ.
🔹 Question 2: What is the fundamental period of the sine and cosine functions?
A) π
B) 2π
C) π/2
D) 4π
✅ Answer: B) 2π
3. Asymptotes in the Tangent Function
Ravi observes that the tangent function has vertical asymptotes at x=π/2, 3π/2, 5π/2.
🔹 Question 3: What is the reason for these vertical asymptotes in the tangent function?
A) The function is undefined at these points
B) The function has a maximum value
C) The function has an amplitude of 2
D) The function crosses the x-axis
✅ Answer: A) The function is undefined at these points
4. Phase Shift in Cosine Function
Ravi notices that the cosine function appears to be a shifted version of the sine function.
🔹 Question 4: By how much is the cosine function shifted relative to the sine function?
A) π
B) π/2 units to the right
C) π/2 units to the left
D) No shift
✅ Answer: B) π/2 units to the right
Conclusion:
By analyzing the graphs of sine, cosine, and tangent functions, Ravi understands the key properties such as amplitude, period, asymptotes, and phase shifts, which are crucial for solving JEE problems.
Case Study: Analyzing Trigonometry Graphs in Real-Life Applications
A city is planning to install a new Ferris wheel at its amusement park. The Ferris wheel has a radius of 30 meters and completes one full rotation every 10 minutes. The height of a passenger above the ground can be modeled using trigonometric functions. The engineers need to analyze the height of a passenger over time to ensure safety and optimize the ride experience.
Objectives:
- Model the height of a passenger using sine, cosine, and tangent functions.
- Analyze the amplitude, period, and phase shift of the graphs.
- Determine the maximum and minimum heights of the passenger.
- Identify key points on the graph to ensure safety and smooth operation.
Data:
- Radius of Ferris wheel (amplitude): 30 meters
- Time for one full rotation (period): 10 minutes
- The lowest point of the Ferris wheel is 5 meters above the ground.
Questions:
1. Modeling the Height Using Sine and Cosine Functions
a) Write the equation for the height h(t)h(t) of a passenger as a function of time tt using the sine function.
b) Write the equation for the height h(t)h(t) of a passenger as a function of time tt using the cosine function.
c) Explain the relationship between the sine and cosine models in this scenario.
Answers:
a) h(t)=30sin(2π10t)+35h(t)=30sin(102πt)+35
b) h(t)=30cos(2π10t−π2)+35h(t)=30cos(102πt−2π)+35
c) The cosine function is a phase-shifted version of the sine function. In this case, the cosine graph is shifted π22π radians (or 2.5 minutes) to the right compared to the sine graph.
2. Analyzing the Graph
a) What is the amplitude of the height function, and what does it represent?
b) What is the period of the height function, and what does it represent?
c) What is the vertical shift of the height function, and what does it represent?
Answers:
a) The amplitude is 30 meters, which represents the radius of the Ferris wheel.
b) The period is 10 minutes, which represents the time for one full rotation of the Ferris wheel.
c) The vertical shift is 35 meters, which represents the height of the center of the Ferris wheel above the ground (5 meters above the lowest point).
3. Maximum and Minimum Heights
a) What is the maximum height of a passenger above the ground?
b) What is the minimum height of a passenger above the ground?
Answers:
a) Maximum height = Amplitude + Vertical shift = 30+35=6530+35=65 meters.
b) Minimum height = Vertical shift – Amplitude = 35−30=535−30=5 meters.
4. Key Points on the Graph
a) At what time(s) does the passenger reach the maximum height?
b) At what time(s) does the passenger reach the minimum height?
c) What is the height of the passenger after 2.5 minutes?
Answers:
a) The passenger reaches the maximum height at t=2.5t=2.5 minutes (using the sine function) or t=5t=5 minutes (using the cosine function).
b) The passenger reaches the minimum height at t=7.5t=7.5 minutes (using the sine function) or t=0t=0 minutes (using the cosine function).
c) After 2.5 minutes, the height is 35 meters (using the sine function).
5. Tangent Function Application
a) Why is the tangent function not suitable for modeling the height of the passenger in this scenario?
b) In what real-life scenario could the tangent function be used to model a situation?
Answers:
a) The tangent function has vertical asymptotes and an unbounded range, which does not match the periodic and bounded nature of the Ferris wheel’s motion.
b) The tangent function could be used to model situations with unbounded growth or decay, such as the angle of elevation of the sun over time.
Conclusion:
By analyzing the graphs of sine and cosine functions, the engineers can accurately model the height of a passenger on the Ferris wheel over time. This information ensures the ride operates safely and provides an enjoyable experience for passengers. The tangent function, while not applicable here, is useful in other real-life scenarios involving unbounded changes.
Assertion & Reason Questions – Trigonometry Graphs for Sine, Cosine & Tangent Functions
Q1.
Assertion (A): The sine and cosine functions have the same amplitude but different phase shifts.
Reason (R): The cosine function is a phase-shifted version of the sine function, shifted by π/2 to the left.
A) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
B) Both Assertion and Reason are true, but Reason is not the correct explanation of Assertion.
C) Assertion is true, but Reason is false.
D) Assertion is false, but Reason is true.
✅ Answer: A) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
Q2.
Assertion (A): The function y=tan(x) has vertical asymptotes at x=π/2,3π/2,5π/2, etc.
Reason (R): The tangent function is undefined where the cosine function is zero.
A) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
B) Both Assertion and Reason are true, but Reason is not the correct explanation of Assertion.
C) Assertion is true, but Reason is false.
D) Assertion is false, but Reason is true.
✅ Answer: A) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
Q3.
Assertion (A): The period of the sine and cosine functions is 2π, while the period of the tangent function is π.
Reason (R): The sine and cosine functions repeat their values every 2π2\pi2π, whereas the tangent function completes one cycle in π\piπ because it has asymptotes at x=π/2,3π/2,…
A) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
B) Both Assertion and Reason are true, but Reason is not the correct explanation of Assertion.
C) Assertion is true, but Reason is false.
D) Assertion is false, but Reason is true.
✅ Answer: A) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
Q4.
Assertion (A): The sine and cosine functions are odd functions, meaning they satisfy f(−x)=−f(x).
Reason (R): The sine function is an odd function, but the cosine function is an even function.
A) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
B) Both Assertion and Reason are true, but Reason is not the correct explanation of Assertion.
C) Assertion is true, but Reason is false.
D) Assertion is false, but Reason is true.
✅ Answer: D) Assertion is false, but Reason is true.
Q5.
Assertion (A): The graph of y=sin(x) is symmetrical about the origin.
Reason (R): The sine function is an odd function, satisfying sin(−x)=−sin(x).
A) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
B) Both Assertion and Reason are true, but Reason is not the correct explanation of Assertion.
C) Assertion is true, but Reason is false.
D) Assertion is false, but Reason is true.
✅ Answer: A) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
Practice MCQs Trigonometry Graphs – Sine, Cosine & Tangent Functions
Question 1:
The graph of y=sin(x) intersects the x-axis at which of the following points?
(a) (0,0) and (π,0)
(b) (0,0) and (π/2,0)
(c) (π/2,0) and (π,0)
(d) (π,0) and (3π/2,0)
Answer: (a)
Question 2:
What is the maximum value of the function y=2cos(x)−1?
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (a)
Question 3:
The graph of y=tan(x) has vertical asymptotes at which of the following values of x?
(a) x=nπ, where n is an integer
(b) x=nπ/2, where n is an odd integer
(c) x=π/2+nπ, where n is an integer
(d) x=π+nπ, where n is an integer
Answer: (c)
Question 4:
Which of the following functions has a period of π?
(a) y=sin(x)
(b) y=cos(2x)
(c) y=tan(x)
(d) y=sin(2x)
Answer: (c)
Question 5:
The graph of y=sin(x) is shifted to the left by 2π units. Which of the following is the equation of the resulting graph?
(a) y=sin(x+π/2)
(b) y=sin(x−π/2)
(c) y=sin(x)+π/2
(d) y=sin(x)−π/2
Answer: (a)
Question 6:
What is the range of the function y=−cos(x)+2?
(a) [−1,1]
(b) [−2,2]
(c) [1,3]
(d) [0,4]
Answer: (c)
Question 7:
The graph of y=tan(x) has x-intercepts at which of the following values of x?
(a) x=nπ, where n is an integer
(b) x=nπ/2, where n is an odd integer
(c) x=π/2+nπ, where n is an integer
(d) x=π+nπ, where n is an integer
Answer: (a)
Question 8:
Which of the following functions is decreasing on the interval (0,π/2)?
(a) y=sin(x) (b) y=cos(x) (c) y=tan(x) (d) y=−sin(x)
Answer: (b)
Question 9:
The graph of y=cos(x) is reflected across the x-axis. Which of the following is the equation of the resulting graph?
(a) y=cos(−x)
(b) y=−cos(x)
(c) y=cos(x)
(d) y=−cos(−x)
Answer: (b)
Question 10:
What is the amplitude of the function y=3sin(2x)+1?
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (c)
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- Graphical Representation of Trigonometric Functions
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