Tamilnadu State Board New Syllabus Samacheer Kalvi 12th Maths Guide Pdf Chapter 4 Inverse Trigonometric Functions Ex 4.2 Textbook Questions and Answers, Notes.

## Tamilnadu Samacheer Kalvi 12th Maths Solutions Chapter 4 Inverse Trigonometric Functions Ex 4.2

Question 1.

Find all values of x such that

(j) -6π ≤ x ≤ 6π and cos x = 0

(ii) -5π ≤ x ≤ 5π and cos x = 1

Solution:

(i) cos x = 0

cos x = cos \(\frac {π}{2}\)

x = (2n + 1) \(\frac {π}{2}\), n = 0, ±1, ±2, ±3, ±4, ±5, -6

(ii) cos x = -1

cos x = cos π

x = (2n + 1) π, n = 0, ±1, ±2, -3

Question 2.

state the reason for cos ^{-1} [cos(\(\frac {-π}{6}\))] ≠ –\(\frac {π}{6}\)

Solution:

cos ^{-1} [cos(-\(\frac {π}{6}\))] = cos ^{-1}[ \(\frac {π}{6}\) ] = \(\frac {π}{6}\) ≠ \(\frac {-π}{6}\) ∉ [0, π]

Which is the principle domain of cosine function [∵ cos(-θ) = cos θ]

Question 3.

Is cos^{-1} (-x) = π – cos^{-1} true? justify your answer.

solution:

Question 4.

Find the principle value of cos^{-1}(\(\frac {1}{2}\))

solution:

y = cos^{-1}(\(\frac {1}{2}\))

cos^{-1}x range is [0, π]

cos y = \(\frac {1}{2}\) = cos = \(\frac {π}{3}\)

y = \(\frac {π}{3}\) ∈ [0, π]

principle value is \(\frac {π}{3}\)

Question 5.

find the value of

(i) 2 cos^{-1} (\(\frac {1}{2}\)) + sin^{-1} (\(\frac {1}{2}\))

(ii) cos^{-1} (\(\frac {1}{2}\)) + sin^{-1}(-1)

(iii) cos^{-1} (cos\(\frac {π}{2}\)cos\(\frac {π}{17}\) – sin\(\frac {π}{7}\)sin\(\frac {π}{17}\))

Solution:

(i) 2 cos^{-1} \(\frac {1}{2}\) + sin^{-1} \(\frac {1}{2}\)

(ii) cos^{-1} \(\frac {1}{2}\) + sin^{-1}(-1)

Question 6.

Find the domain of

(i) f(x) = sin^{-1} (\(\frac {|x|-2}{3}\)) + cos^{-1} (\(\frac {1-|x|}{4}\))

(ii) g(x) = sin^{-1} x + cos^{-1} x

Solution:

(i) -1 ≤ sin^{-1} (x) ≤ 1

-1 ≤ \(\frac {|x|-2}{3}\) ≤ 1

-3 ≤ |x| – 2 ≤ 3

-3 + 2 ≤ |x| ≤ 3 + 2

-1 ≤ |x| ≤ 5

|x| ≤ 5

since -1 ≤ |x| is not possible

-5 ≤ x ≤ 5 ………. (1)

By the definitions

-1 ≤ cos^{-1} (x) ≤ 1

-1 ≤ \(\frac {1-|x|}{4}\) ≤ 1

-4 ≤ 1 – |x| ≤ 4

-5 ≤ -|x| ≤ 3

-3 ≤ |x| ≤ 5

-3 ≤ |x| is not possible

-5 ≤ x ≤ 5 ………. (2)

From 1 and 2 we get

domain is x ∈ [-5, 5]

(ii) g(x) = sin^{-1} x + cos^{-1} x

range of sin x and cos x is [-1, 1]

-1 ≤ x ≤ 1

∴ x ∈ [-1, 1]

The domain of g(x) = [-1, 1]

Question 7.

For what value of x, the inequality

\(\frac {π}{2}\) < cos^{-1} (3x – 1) < π holds?

Solution:

\(\frac {π}{2}\) < cos^{-1} (3x – 1) < π

cos \(\frac {π}{2}\) < (3x – 1) < cos π

0 < 3x < 1 < -1

0 + 1 < 3x < -1 + 1

1 < 3x < 0

The inequality is true, only when

0 < x < \(\frac {1}{3}\)

Question 8.

Find the value of

(i) cos[cos^{-1}(\(\frac {4}{5}\)) + sin^{-1}(\(\frac {4}{5}\))]

(ii) (cos^{-1}(cos \(\frac {4π}{3}\))) + cos^{-1}(cos(\(\frac {5π}{4}\)))

Solution: