Graphs Of Trigonometric Functions
Class 11th Mathematics RD Sharma Solution
- f(x) = 2 sin x, 0 ≤ x ≤ π Sketch the graphs of the following functions :…
- g(x) = 3 sin (x - pi /4) , 0 less than equal to x less than equal to 5 pi /4…
- h(x) = 2 sin 3x, 0 ≤ x ≤ 2 π/3 Sketch the graphs of the following functions :…
- phi (x) = 2sin (2x - pi /3) , 0 less than equal to x less than equal to 7 pi /5…
- psi (x) = 4sin3 (x - pi /4) , 0 less than equal to x less than equal to 2 pi…
- theta (x) = sin (x/2 - pi /4) , 0 less than equal to x less than equal to 4 pi…
- u(x) = sin^2 x, 0 ≤ x ≤ 2π υ(x) = |sin x|, 0 ≤ x ≤ 2π Sketch the graphs of the…
- f(x) = 2 sin πx, 0 ≤ x ≤ 2. Sketch the graphs of the following functions :…
- f(x) = sin x, g(x) = sin (x + pi /4) Sketch the graphs of the following pairs…
- f(x) = sin x, g(x) = sin 2x Sketch the graphs of the following pairs of…
- f(x) = sin 2x, g(x) = 2 sin x Sketch the graphs of the following pairs of…
- f(x) = sin x/2 , g(x) = sin x Sketch the graphs of the following pairs of…
- f(x) = cos (x - pi /4) Sketch the graphs of the following trigonometric…
- g(x) = cos (x + pi /4) Sketch the graphs of the following trigonometric…
- h(x) = cos^2 2x Sketch the graphs of the following trigonometric functions :…
- phi (x) = 2cos (x - pi /6) Sketch the graphs of the following trigonometric…
- psi (x) = cos3x Sketch the graphs of the following trigonometric functions :…
- u (x) = cos^2 x/2 Sketch the graphs of the following trigonometric functions :…
- f(x) = cos π x Sketch the graphs of the following trigonometric functions :…
- g(x) = cos 2π x Sketch the graphs of the following trigonometric functions :…
- y = cos x and y = cos (x - pi /4) Sketch the graphs of the following curves on…
- y = cos 2x and y = cos (x - pi /4) Sketch the graphs of the following curves on…
- y = cos x and y = cos x/2 Sketch the graphs of the following curves on the same…
- y = cos^2 x and y = cos x Sketch the graphs of the following curves on the same…
- f(x) = 2 cosec πx Sketch the graphs of the following functions :
- f(x) = 3 sec x Sketch the graphs of the following functions :
- f(x) = cot 2x Sketch the graphs of the following functions :
- f(x) = 2sec πx Sketch the graphs of the following functions :
- f(x) = tan^2 x Sketch the graphs of the following functions :
- f(x) = cot^2 x Sketch the graphs of the following functions :
- f (x) = cot pi x/2 Sketch the graphs of the following functions :…
- f(x) = sec^2 x Sketch the graphs of the following functions :
- f(x) = cosec^2 x Sketch the graphs of the following functions :
- f(x) = tan 2x Sketch the graphs of the following functions :
Exercise 6.1
Question 1.Sketch the graphs of the following functions :
f(x) = 2 sin x, 0 ≤ x ≤ π
Answer:
We know that g (x) = sin x is a periodic function with period π.
∴ f (x) = 2 sin x is a periodic function with period π. So, we will draw the graph of f (x) = 2 sin x in the interval [0, π]. The values of f (x) = 2 sin x at various points in [0, π] are listed in the following table:
By plotting the above points, we obtain the required curve.
Question 2.
Sketch the graphs of the following functions :
g(x) = 3 sin 
Answer:
We know that if f(x) is a periodic function with period T, then f (ax + b) is periodic with period T/|a|.
∴ 
is a periodic function with period π. So, we will draw the graph of in the interval [0, 5π/4]. The values of at various points in [0, 5π/4] are listed in the following table:
By plotting the above points, we obtain the required curve.
Question 3.
Sketch the graphs of the following functions :
h(x) = 2 sin 3x, 0 ≤ x ≤ 2 π/3
Answer:
We know that g (x) = sin x is a periodic function with period 2π.
∴ h (x) = 2 sin 3x is a periodic function with period 2π/3. So, we will draw the graph of h (x) = 2 sin 3x in the interval [0, 2π/3]. The values of h (x) = 2 sin 3x at various points in [0, 2π/3] are listed in the following table:
By plotting the above points, we obtain the required curve.
Question 4.
Sketch the graphs of the following functions :
Answer:
We know that if f(x) is a periodic function with period T, then f (ax + b) is periodic with period T/|a|.
∴ 
is a periodic function with period π. So, we will draw the graph of in the interval [0, 7π/5]. The values of at various points in [0, 7π/5] are listed in the following table:
By plotting the above points, we obtain the required curve.
Question 5.
Sketch the graphs of the following functions :
Answer:
We know that if f(x) is a periodic function with period T, then f (ax + b) is periodic with period T/|a|.
∴ 
is a periodic function with period 2π. So, we will draw the graph of in the interval [0, 2π]. The values of at various points in [0, 2π] are listed in the following table:
By plotting the above points, we obtain the required curve.
Question 6.
Sketch the graphs of the following functions :
Answer:
We know that if f(x) is a periodic function with period T, then f (ax + b) is periodic with period T/|a|.
∴ 
is a periodic function with period 4π. So, we will draw the graph of in the interval [0, 4π]. The values of at various points in [0, 4π] are listed in the following table:
By plotting the above points, we obtain the required curve.
Question 7.
Sketch the graphs of the following functions :
u(x) = sin2 x, 0 ≤ x ≤ 2π υ(x) = |sin x|, 0 ≤ x ≤ 2π
Answer:
We know that g (x) = sin x is a periodic function with period π.
∴ u (x) = sin2 x is a periodic function with period 2π. So, we will draw the graph of u (x) = sin2 x in the interval [0, 2π]. The values of u (x) = sin2 x at various points in [0, 2π] are listed in the following table:
By plotting the above points, we obtain the required curve.
Then,
∴ u (x) = |sin x| is a periodic function with period 2π. So, we will draw the graph of u (x) = |sin x| in the interval [0, 2π]. The values of u (x) = |sin x| at various points in [0, 2π] are listed in the following table:
By plotting the above points, we obtain the required curve.
Question 8.
Sketch the graphs of the following functions :
f(x) = 2 sin πx, 0 ≤ x ≤ 2.
Answer:
We know that g (x) = sin x is a periodic function with period 2π.
∴ f (x) = 2 sin πx is a periodic function with period 2. So, we will draw the graph of f (x) = 2 sin πx in the interval [0, 2]. The values of f (x) = 2 sin πx at various points in [0, 2] are listed in the following table:
By plotting the above points, we obtain the required curve.
Question 9.
Sketch the graphs of the following pairs of functions on the same axes :
f(x) = sin x, g(x) = sin 
Answer:
We observe that the functions f(x) = sin x and g (x) = sin (x + π/4) are periodic functions with periods 2π and 7π/4.
The values of these functions are tabulated below:
Values of f (x) = sin x in [0, 2π]
Values of g (x) = sin (x + π/4) in [0, 7π/4]
By plotting the above points, we obtain the required curve.
Question 10.
Sketch the graphs of the following pairs of functions on the same axes :
f(x) = sin x, g(x) = sin 2x
Answer:
We observe that the functions f(x) = sin x and g (x) = sin 2x are periodic functions with periods 2π and π.
The values of these functions are tabulated below:
Values of f (x) = sin x in [0, 2π]
Values of g (x) = sin (2x) in [0, π]
By plotting the above points, we obtain the required curve.
Question 11.
Sketch the graphs of the following pairs of functions on the same axes :
f(x) = sin 2x, g(x) = 2 sin x
Answer:
We observe that the functions f(x) = sin 2x and g (x) = 2 sin x are periodic functions with periods π and π.
The values of these functions are tabulated below:
Values of f (x) = sin (2x) in [0, π]
Values of g (x) = 2 sin x in [0, π]
By plotting the above points, we obtain the required curve.
Question 12.
Sketch the graphs of the following pairs of functions on the same axes :
f(x) = sin
, g(x) = sin x
Answer:
We observe that the functions f(x) = sin x/2 and g (x) = sin x are periodic functions with periods π and 2π.
The values of these functions are tabulated below:
Values of f (x) = sin x/2 in [0, π]
Values of g (x) = sin (x) in [0, 2π]
By plotting the above points, we obtain the required curve.
Exercise 6.2
Question 1.Sketch the graphs of the following trigonometric functions :
f(x) = cos 
Answer:
We know that g (x) = cos x is a periodic function with period 2π.
∴ f (x) = cos (x - π/4) is a periodic function with period π. So, we will draw the graph of f (x) = cos (x - π/4) in the interval [0, π]. The values of f (x) = cos (x - π/4) at various points in [0, π] are listed in the following table:
By plotting the above points, we obtain the required curve.
Question 2.
Sketch the graphs of the following trigonometric functions :
g(x) = cos
Answer:
We know that f (x) = cos x is a periodic function with period 2π.
∴ g (x) = cos (x + π/4) is a periodic function with period π. So, we will draw the graph of g (x) = cos (x + π/4) in the interval [0, π]. The values of g (x) = cos (x + π/4) at various points in [0, π] are listed in the following table:
By plotting the above points, we obtain the required curve.
Question 3.
Sketch the graphs of the following trigonometric functions :
h(x) = cos2 2x
Answer:
We know that f (x) = cos x is a periodic function with period 2π.
∴ h (x) = cos2 2x is a periodic function with period π. So, we will draw the graph of h (x) = cos2 2x in the interval [0, π]. The values of h (x) = cos2 2x at various points in [0, π] are listed in the following table:
By plotting the above points, we obtain the required curve.
Question 4.
Sketch the graphs of the following trigonometric functions :
Answer:
We know that f (x) = cos x is a periodic function with period 2π.
∴ ϕ (x) = 2cos (x - π/6) is a periodic function with period π. So, we will draw the graph of ϕ (x) = 2cos (x - π/6) in the interval [0, π]. The values of ϕ (x) = 2cos (x - π/6) at various points in [0, π] are listed in the following table:
By plotting the above points, we obtain the required curve.
Question 5.
Sketch the graphs of the following trigonometric functions :
Answer:
We know that f (x) = cos x is a periodic function with period 2π.
∴ ψ (x) = cos (3x) is a periodic function with period 2π/3. So, we will draw the graph of ψ (x) = cos (3x) in the interval [0, 2π/3]. The values of ψ (x) = cos (3x) at various points in [0, 2π/3] are listed in the following table:
By plotting the above points, we obtain the required curve.
Question 6.
Sketch the graphs of the following trigonometric functions :
Answer:
We know that f (x) = cos x is a periodic function with period 2π.
∴ u (x) = cos2 (x/2) is a periodic function with period π. So, we will draw the graph of u (x) = cos2 (x/2) in the interval [0, π]. The values of u (x) = cos2 (x/2) at various points in [0, π] are listed in the following table:
By plotting the above points, we obtain the required curve.
Question 7.
Sketch the graphs of the following trigonometric functions :
f(x) = cos π x
Answer:
We know that g (x) = cos x is a periodic function with period 2π.
∴ f (x) = cos (πx) is a periodic function with period 2. So, we will draw the graph of f (x) = cos (πx) in the interval [0, 2]. The values of f (x) = cos (πx) at various points in [0, 2] are listed in the following table:
By plotting the above points, we obtain the required curve.
Question 8.
Sketch the graphs of the following trigonometric functions :
g(x) = cos 2π x
Answer:
We know that f (x) = cos x is a periodic function with period 2π.
∴ g (x) = cos (2πx) is a periodic function with period 1. So, we will draw the graph of g (x) = cos (2πx) in the interval [0, 1]. The values of g (x) = cos (2πx) at various points in [0, 1] are listed in the following table:
By plotting the above points, we obtain the required curve.
Question 9.
Sketch the graphs of the following curves on the same scale and the same axes :
y = cos x and y = cos 
Answer:
We observe that the functions y = cos x and y = cos (x - π/4) are periodic functions with periods π and π.
The values of these functions are tabulated below:
By plotting the above points, we obtain the required curve.
Question 10.
Sketch the graphs of the following curves on the same scale and the same axes :
y = cos 2x and y = cos
Answer:
We observe that the functions y = cos 2x and y = cos 2(x - π/4) are periodic functions with periods π and π.
The values of these functions are tabulated below:
By plotting the above points, we obtain the required curve.
Question 11.
Sketch the graphs of the following curves on the same scale and the same axes :
y = cos x and y = cos 
Answer:
We observe that the functions y = cos x and y = cos (x/2) are periodic functions with periods π and π.
The values of these functions are tabulated below:
By plotting the above points, we obtain the required curve.
Question 12.
Sketch the graphs of the following curves on the same scale and the same axes :
y = cos2 x and y = cos x
Answer:
We observe that the functions y = cos2 x and y = cos (x) are periodic functions with period 2π.
The values of these functions are tabulated below:
By plotting the above points, we obtain the required curve.
Exercise 6.3
Question 1.Sketch the graphs of the following functions :
f(x) = 2 cosec πx
Answer:
We know that g (x) = cosec x is a periodic function with period 2π.
∴ f (x) = 2 cosec (πx) is a periodic function with period 2. So, we will draw the graph of f (x) = 2 cosec (πx) in the interval [0, 2]. The values of f (x) = 2 cosec (πx) at various points in [0, 2] are listed in the following table:
By plotting the above points, we obtain the required curve.
Question 2.
Sketch the graphs of the following functions :
f(x) = 3 sec x
Answer:
We know that g (x) = sec x is a periodic function with period π.
∴ f (x) = 3 sec (x) is a periodic function with period π. So, we will draw the graph of f (x) = 3 sec (x) in the interval [0, π]. The values of f (x) = 3 sec (x) at various points in [0, π] are listed in the following table:
By plotting the above points, we obtain the required curve.
Question 3.
Sketch the graphs of the following functions :
f(x) = cot 2x
Answer:
We know that g (x) = cot x is a periodic function with period π.
∴ f (x) = cot (2x) is a periodic function with period π. So, we will draw the graph of f (x) = cot (2x) in the interval [0, π]. The values of f (x) = cot (2x) at various points in [0, π] are listed in the following table:
By plotting the above points, we obtain the required curve.
Question 4.
Sketch the graphs of the following functions :
f(x) = 2sec πx
Answer:
We know that g (x) = sec x is a periodic function with period π.
∴ f (x) = 2 sec (πx) is a periodic function with period 1. So, we will draw the graph of f (x) = 2 sec (πx) in the interval [0, 1]. The values of f (x) = 2 sec (πx) at various points in [0, 1] are listed in the following table:
By plotting the above points, we obtain the required curve.
Question 5.
Sketch the graphs of the following functions :
f(x) = tan2 x
Answer:
We know that g (x) = tan x is a periodic function with period π.
∴ f (x) = tan2 (x) is a periodic function with period π. So, we will draw the graph of f (x) = tan2 (x) in the interval [0, π]. The values of f (x) = tan2 (x) at various points in [0, π] are listed in the following table:
By plotting the above points, we obtain the required curve.
Question 6.
Sketch the graphs of the following functions :
f(x) = cot2 x
Answer:
We know that g (x) = cot x is a periodic function with period π.
∴ f (x) = cot2 (x) is a periodic function with period π. So, we will draw the graph of f (x) = cot2 (x) in the interval [0, π]. The values of f (x) = cot2 (x) at various points in [0, π] are listed in the following table:
By plotting the above points, we obtain the required curve.
Question 7.
Sketch the graphs of the following functions :
Answer:
We know that g (x) = cot x is a periodic function with period π.
∴ f (x) = cot (πx/2) is a periodic function with period 2. So, we will draw the graph of f (x) = cot (πx/2) in the interval [0, 2]. The values of f (x) = cot (πx/2) at various points in [0, 2] is listed in the following table:
By plotting the above points, we obtain the required curve.
Question 8.
Sketch the graphs of the following functions :
f(x) = sec2 x
Answer:
We know that g (x) = sec x is a periodic function with period π.
∴ f (x) = sec2 (x) is a periodic function with period π. So, we will draw the graph of f (x) = sec2 (x) in the interval [0, π]. The values of f (x) = sec2 (x) at various points in [0, π] are listed in the following table:
By plotting the above points, we obtain the required curve.
Question 9.
Sketch the graphs of the following functions :
f(x) = cosec2 x
Answer:
We know that g (x) = cosec x is a periodic function with period 2π.
∴ f (x) = cosec2 (x) is a periodic function with period 2π. So, we will draw the graph of f (x) = cosec2 (x) in the interval [0, 2π]. The values of f (x) = cosec2 (x) at various points in [0, 2π] are listed in the following table:
By plotting the above points, we obtain the required curve.
Question 10.
Sketch the graphs of the following functions :
f(x) = tan 2x
Answer:
We know that g (x) = tan x is a periodic function with period π.
∴ f (x) = tan (2x) is a periodic function with period π/2. So, we will draw the graph of f (x) = tan (2x) in the interval [0, π/2]. The values of f (x) = tan (2x) at various points in [0, π/2] are listed in the following table:
By plotting the above points, we obtain the required curve.
