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 $$\frac{\pi}{|5|}=\frac{\pi}{5}$$

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

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:

Question 2.
cos 2x in [0, π]
Solution:

Question 3.
sin 2x in the interval (0, π)
Solution:

Question 4.
sin x in the interval [-π, +π]
Solution:

Question 5.
cos2x in [0, π]
Solution:

III.

Question 1.
Sketch the region enclosed by y = sin x, y = cos x and X-axis in the interval [0, π].
Solution: