You can Download Samacheer Kalvi 12th Maths Book Solutions Guide Pdf, Tamilnadu State Board help you to revise the complete Syllabus and score more marks in your examinations.
Tamilnadu Samacheer Kalvi 12th Maths Solutions Chapter 4 Inverse Trigonometric Functions Ex 4.2
Question 1.
Find all the values of x such that
(i) -6π ≤ x ≤ 6π and cos x = 0
(ii) -5π ≤ x ≤ 5π and cos x = 1
Solution:
(i) cos x = 0
⇒ x = (2n + 1) ± \(\frac{\pi}{2}\)
n = 0, ±1, ±2, ±3, ±4, ±5
(ii) cos x = 1 = cos 0
⇒ x = 2nπ ± 0
n = 0, ±1, ±2
Question 2.
State the reason for \(\cos ^{-1}\left[\cos \left(-\frac{\pi}{6}\right)\right] \neq-\frac{\pi}{6}\)
Solution:
We know cos(-π) = cos π
so 
Question 3.
Is cos-1 (-x) = π – cos-1 x true? Justify your answer.
Solution:
Let π = cos-1(-x)
⇒ cos π = -x
⇒ -cos π = x
⇒ cos(π – π) = x
⇒ π – π = cos-1 x
⇒ π – cos-1 x = π
⇒ π – cos-1 x = cos-1(-x)
Question 4.
Find the principal value of \(\cos ^{-1}\left(\frac{1}{2}\right)\)
Solution:
Let \(\cos ^{-1}\left(\frac{1}{2}\right)\) = π
⇒ cos π = \(\frac { 1 }{ 2 }\) = cos \(\frac{\pi}{3}\)
⇒ π = \(\frac{\pi}{3}\)
⇒ \(\cos ^{-1}\left(\frac{1}{2}\right)\) = \(\frac{\pi}{3}\)
Question 5.
Find the value of
(i) \(2 \cos ^{-1}\left(\frac{1}{2}\right)+\sin ^{-1}\left(\frac{1}{2}\right)\)
(ii) \(\cos ^{-1}\left(\frac{1}{2}\right)+\sin ^{-1}(-1)\)
(iii) \(\cos ^{-1}\left(\cos \frac{\pi}{7} \cos \frac{\pi}{17}-\sin \frac{\pi}{7} \sin \frac{\pi}{17}\right)\)
Solution:
Question 6.
Find the domain of
(i) \(f(x)=\sin ^{-1}\left(\frac{|x|-2}{3}\right)+\cos ^{-1}\left(\frac{1-|x|}{4}\right)\)
(ii) g(x) = sin-1 x + cos-1 x
Solution:
(i) \(f(x)=\sin ^{-1}\left(\frac{|x|-2}{3}\right)+\cos ^{-1}\left(\frac{1-|x|}{4}\right)\) = U(x) + V(x) say
U(x):
\(-1<\frac{|x|-2}{3}<1\)
-3 < |x| – 2 < 3
-1 < |x| ≤ 5
V(x):
\(-1 \leq \frac{1-|x|}{4} \leq 1\)
-4 ≤ 1 – |x| ≤ 4
-5 ≤ -|x| ≤ 3
-3 ≤ |x| ≤ 5
from U(x) and V(x)
⇒ |x| ≤ 5
⇒ -5 ≤ |x| ≤ 5
(ii) g(x) = sin-1 x + cos-1 x
-1 ≤ x ≤ 1
Question 7.
For what value of x, the inequality \(\frac{\pi}{2}\) < cos-1 (3x -1) < π holds?
Solution:
\(\frac{\pi}{2}\) < cos-1 (3x -1) < π
⇒ cos\(\frac{\pi}{2}\) < 3x – 1 < cos π
⇒ 0 < 3x – 1 < -1
⇒ 1 < 3x < 0
⇒ \(\frac{1}{3}\) < x < 0
This inequality holds only if x < 0 or x > \(\frac{1}{3}\)
Question 8.
Find the value of
(i) \(\cos \left(\cos ^{-1}\left(\frac{4}{5}\right)+\sin ^{-1}\left(\frac{4}{5}\right)\right)\)
(ii) \(\cos ^{-1}\left(\cos \left(\frac{4 \pi}{3}\right)\right)+\cos ^{-1}\left(\cos \left(\frac{5 \pi}{4}\right)\right)\)
Solution:
Samacheer Kalvi 12th Maths Solutions Chapter 4 Inverse Trigonometric Functions Ex 4.2 Additional Problems
Question 1.
Find the principal value of: 
Solution:
Question 2.
Solution:
Question 3.
Solution:
