Students often turn to ML Aggarwal Maths for Class 11 Solutions Chapter 3 Trigonometry Ex 3.4 to clarify doubts and improve problem-solving skills.

## ML Aggarwal Class 11 Maths Solutions Section A Chapter 3 Trigonometry Ex 3.4

Question 1.
Draw the graphs of the following functions :
(i) sin 3x
(ii) 3 sin x
(iii) 2 sin 2x.
Also write their range and period.
Solution:
(i) Given y = sin 3x
= sin (3x + 2π)
= sin 3 (x + $$\frac{2 \pi}{3}$$)
∴ The period of sin 3x is $$\frac{2 \pi}{3}$$.
So it is suffices to draw the graph in the interval 0 to $$\frac{2 \pi}{3}$$ and repeat it over other intervals.
We construct the table of values are given as under:

A portion of graph is given as under
since – 1 ≤ sin 3x ≤ 1
∴ Range = {y ; y ∈ R : – 1 ≤ y ≤ 1}

(ii) Let y = 3 sin x 3 sin (x + 2π), it is defined for all x ∈ R.
Thus period of 3 sin x be 2π.
So it is sufficient to draw the graph in the interval 0 to 2π and repeat it over other intervals.
Now construct the table of values is as under :

A portion of graph is given as under :

since, – 1 ≤ sin x ≤ 1 ∀ x ∈ R
⇒ – 3 ≤ 3 sin x ≤ 3
Thus, Rf = {y ; y ∈ R, – 3 ≤ y ≤ 3}
= [- 3, 3]

(iii) Let y = 2 sin 2x. it is defined for all x ∈ R.
⇒ y = 2 sin (2x + 2π)
= 2 sin 2 (x + π)
Thus period of 2 sin 2x be π.
So it is sufficient to draw the graph in the interval 0 to π and repeat it over other intervals.
Now construct the table of values is as under:

The portion of graph is represented as under :

since – 1 ≤ sin 2x ≤ 1 ∀ x ∈ R
⇒ – 2 ≤ y ≤ 2 ∀ x ∈ R
∴ range of f (x) = Rf
= {y : y ∈ R – 2 ≤ y ≤ 2} = [- 2, 2]

Question 2.
Draw the graph or the following functions:
(i) cos $$\frac{x}{2}$$
(ii) 3 cos 2x
(iii) 2 cos 3x
Also write their range and period.
Solution:
(i) Let y = cos $$\frac{x}{2}$$
which is defined for all x ∈ R.
Now y = cos $$\frac{x}{2}$$
= cos ($$\frac{x}{2}$$ + 2π)
= cos $$\frac{1}{2}$$ + (x + 4π)
∴ period of y be 4π.
So it is sufflcient to draw the graph in the interval 0 to 4π and repeat it over other intervals.
We construct the table of values is as under:
A portion of graph is given as under:

since, – 1 ≤ cos $$\frac{x}{2}$$ ≤ 1 ∀ x ∈ R
Thus Rf = [- 1, 1]

(ii) Let y = 3 cos 2x
= 3 cos (2x + 2π)
= 3 cos 2 (x + π), which is defined ∀ x ∈ R.
The period of 3 cos 2x be π.
So it is sufficient to draw the graph in the interval 0 to π and repeat it over other intervals.
We construct the table of values is as under:
The portion of graph is given as under:

since, – 1 ≤ cos 2x ≤ 1 ∀ x ∈ R
⇒ – 3 ≤ 3 cos 2x ≤ 3
⇒ – 3 ≤ y ≤ 3
∴ Range = {y : y ∈ R ; – 3 ≤ y ≤ 3} = [- 3, 3]

(iii) Let y = 2 cos 3x
= 2 cos (3x + 2π)
= 2 cos 3 (x + $$\frac{2 \pi}{3}$$)
which is defined for all x ∈ R.
Thus, the period of 2 cos 3x be $$\frac{2 \pi}{3}$$.
So it is sufficient to draw the graph in the interval 0 to $$\frac{2 \pi}{3}$$ and then repeat it over other intervals.
We construct the table of values is given as under:

The portion of graph is given as under :

since – 1 ≤ cos 3x ≤ 1 ∀ x ∈ R – 2 ≤ 2 cos 3x ≤ 2
∴ Range = [- 2, 2]