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 + …… + n^{2}x) (n any positive integer)

Solution:

tan(1^{2} + 2^{2} + 3^{2} + …… + n^{2}) 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.

cos^{2}x 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: