Practicing the Intermediate 1st Year Maths 1A Textbook Solutions Inter 1st Year Maths 1A Trigonometric Ratios up to Transformations Solutions Exercise 6(b) will help students to clear their doubts quickly.

Intermediate 1st Year Maths 1A Trigonometric Ratios up to Transformations Solutions Exercise 6(b)

I. Find the periods for the given 1 – 5 functions.

Question 1.
cos(3x + 5) + 7
Solution:
f(x) = cos(3x + 5) + 7
We know that the function g(x) = cos x for all x ∈ R has the period 2π.
Now f(x) = cos(3x + 5) + 7
We get that f(x) is periodic and the period of f is \(\frac{2 \pi}{|3|}=\frac{2 \pi}{3}\)

Question 2.
tan 5x
Solution:
The function g(x) = tan x periodic and π is the period.
∴ f(x) = tan 5x periodic and its period is \(\frac{\pi}{|5|}=\frac{\pi}{5}\)

Inter 1st Year Maths 1A Trigonometric Ratios up to Transformations Solutions Ex 6(b)

Question 3.
\(\cos \left(\frac{4 x+9}{5}\right)\)
Solution:
The function h(x) = cos x for all x ∈ R has the period 2π.
Now f(x) = \(\cos \left(\frac{4 x}{5}+\frac{9}{5}\right)\) is periodic and period of f is \(\frac{2 \pi}{\left(\frac{4}{5}\right)}=\frac{5 \pi}{2}\)

Question 4.
|sin x|
Solution:
The function h(x) = sin x for all x ∈ R has the period 2π.
But f(x) = |sin x| is periodic and its period is π.
∵ f(x + π) = |sin(x + π)|
= |-sin x|
= sin x

Question 5.
tan(x + 4x + 9x + …… + n2x) (n any positive integer)
Solution:
tan(12 + 22 + 32 + …… + n2) x = \(\tan \left[\frac{n(n+1)(2 n+1)}{6}\right] x\)
period = \(\frac{6 \pi}{n(n+1)(2 n+1)}\)

Question 6.
Find a sine function whose period is \(\frac{2}{3}\)
Solution:
\(\frac{2 \pi}{|k|}=\frac{2}{3}\)
3π = |k|
∴ sin kx = sin 3πx

Inter 1st Year Maths 1A Trigonometric Ratios up to Transformations Solutions Ex 6(b)

Question 7.
Find a cosine function whose period is 7.
Solution:
\(\frac{2 \pi}{|k|}\) = 7
\(\frac{2 \pi}{7}\) = |k|
∴ cos kx = cos \(\frac{2 \pi}{7}\) x

II. Sketch the graph of the following functions.

Question 1.
tan x between 0 and \(\frac{\pi}{4}\)
Solution:
Inter 1st Year Maths 1A Trigonometric Ratios up to Transformations Solutions Ex 6(b) II Q1

Question 2.
cos 2x in [0, π]
Solution:
Inter 1st Year Maths 1A Trigonometric Ratios up to Transformations Solutions Ex 6(b) II Q2

Question 3.
sin 2x in the interval (0, π)
Solution:
Inter 1st Year Maths 1A Trigonometric Ratios up to Transformations Solutions Ex 6(b) II Q3

Inter 1st Year Maths 1A Trigonometric Ratios up to Transformations Solutions Ex 6(b)

Question 4.
sin x in the interval [-π, +π]
Solution:
Inter 1st Year Maths 1A Trigonometric Ratios up to Transformations Solutions Ex 6(b) II Q4

Question 5.
cos2x in [0, π]
Solution:
Inter 1st Year Maths 1A Trigonometric Ratios up to Transformations Solutions Ex 6(b) II Q5

III.

Question 1.
Sketch the region enclosed by y = sin x, y = cos x and X-axis in the interval [0, π].
Solution:
Inter 1st Year Maths 1A Trigonometric Ratios up to Transformations Solutions Ex 6(b) III Q1