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RD Sharma Class 12 Solutions - Inverse Trigonometric Functions | Mathematics (Maths) Class 12 - JEE PDF Download

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4. Inverse Trigonometric Functions
Exercise 4.1
1 A. Question
Find the principal value of each of the following:
Answer
Let 
Then sin y = 
We know that the principal value of  is 
Therefore the principal value of  is 
1 B. Question
Find the principal value of each of the following:
Answer
Let 
cos y = 
We need to find the value of y.
We know that the value of cos is negative for the second quadrant and hence the value lies in [0, p].
cos y = – cos 
cos y = 
y = 
1 C. Question
Find the principal value of each of the following:
Answer
Page 2


4. Inverse Trigonometric Functions
Exercise 4.1
1 A. Question
Find the principal value of each of the following:
Answer
Let 
Then sin y = 
We know that the principal value of  is 
Therefore the principal value of  is 
1 B. Question
Find the principal value of each of the following:
Answer
Let 
cos y = 
We need to find the value of y.
We know that the value of cos is negative for the second quadrant and hence the value lies in [0, p].
cos y = – cos 
cos y = 
y = 
1 C. Question
Find the principal value of each of the following:
Answer
1 D. Question
Find the principal value of each of the following:
Answer
1 E. Question
Find the principal value of each of the following:
Answer
Let 
Then 
Page 3


4. Inverse Trigonometric Functions
Exercise 4.1
1 A. Question
Find the principal value of each of the following:
Answer
Let 
Then sin y = 
We know that the principal value of  is 
Therefore the principal value of  is 
1 B. Question
Find the principal value of each of the following:
Answer
Let 
cos y = 
We need to find the value of y.
We know that the value of cos is negative for the second quadrant and hence the value lies in [0, p].
cos y = – cos 
cos y = 
y = 
1 C. Question
Find the principal value of each of the following:
Answer
1 D. Question
Find the principal value of each of the following:
Answer
1 E. Question
Find the principal value of each of the following:
Answer
Let 
Then 
We know that the principal value of  is 
Therefore the principal value of  is .
1 F. Question
Find the principal value of each of the following:
Answer
Let y = 
Therefore, sin y = 
We know that the principal value of  is 
And 
Therefore the principal value of  is .
2 A. Question
Find the principal value of each of the following:
Answer
2 B. Question
Find the principal value of each of the following:
Answer
Page 4


4. Inverse Trigonometric Functions
Exercise 4.1
1 A. Question
Find the principal value of each of the following:
Answer
Let 
Then sin y = 
We know that the principal value of  is 
Therefore the principal value of  is 
1 B. Question
Find the principal value of each of the following:
Answer
Let 
cos y = 
We need to find the value of y.
We know that the value of cos is negative for the second quadrant and hence the value lies in [0, p].
cos y = – cos 
cos y = 
y = 
1 C. Question
Find the principal value of each of the following:
Answer
1 D. Question
Find the principal value of each of the following:
Answer
1 E. Question
Find the principal value of each of the following:
Answer
Let 
Then 
We know that the principal value of  is 
Therefore the principal value of  is .
1 F. Question
Find the principal value of each of the following:
Answer
Let y = 
Therefore, sin y = 
We know that the principal value of  is 
And 
Therefore the principal value of  is .
2 A. Question
Find the principal value of each of the following:
Answer
2 B. Question
Find the principal value of each of the following:
Answer
3 A. Question
Find the domain of each of the following functions:
f(x) = sin
–1
x
2
Answer
Domain of  lies in the interval [–1, 1].
Therefore domain of  lies in the interval [–1, 1].
–1  1
But  cannot take negative values,
So, 0 1
–1  1
Hence domain of  is [–1, 1].
3 B. Question
Find the domain of each of the following functions:
f(x) = sin
–1
x + sinx
Answer
Domain of  lies in the interval [–1, 1].
–1  1.
The domain of sin x lies in the interval 
–1.57  1.57
From the above we can see that the domain of sin
–1
x + sinx is the intersection of the domains of sin
–1
x and
sin x.
So domain of sin
–1
x + sinx is [–1, 1].
3 C. Question
Find the domain of each of the following functions:
f(x) = 
Answer
Domain of  lies in the interval [–1, 1].
Therefore, Domain of  lies in the interval [–1, 1].
–1  1
Page 5


4. Inverse Trigonometric Functions
Exercise 4.1
1 A. Question
Find the principal value of each of the following:
Answer
Let 
Then sin y = 
We know that the principal value of  is 
Therefore the principal value of  is 
1 B. Question
Find the principal value of each of the following:
Answer
Let 
cos y = 
We need to find the value of y.
We know that the value of cos is negative for the second quadrant and hence the value lies in [0, p].
cos y = – cos 
cos y = 
y = 
1 C. Question
Find the principal value of each of the following:
Answer
1 D. Question
Find the principal value of each of the following:
Answer
1 E. Question
Find the principal value of each of the following:
Answer
Let 
Then 
We know that the principal value of  is 
Therefore the principal value of  is .
1 F. Question
Find the principal value of each of the following:
Answer
Let y = 
Therefore, sin y = 
We know that the principal value of  is 
And 
Therefore the principal value of  is .
2 A. Question
Find the principal value of each of the following:
Answer
2 B. Question
Find the principal value of each of the following:
Answer
3 A. Question
Find the domain of each of the following functions:
f(x) = sin
–1
x
2
Answer
Domain of  lies in the interval [–1, 1].
Therefore domain of  lies in the interval [–1, 1].
–1  1
But  cannot take negative values,
So, 0 1
–1  1
Hence domain of  is [–1, 1].
3 B. Question
Find the domain of each of the following functions:
f(x) = sin
–1
x + sinx
Answer
Domain of  lies in the interval [–1, 1].
–1  1.
The domain of sin x lies in the interval 
–1.57  1.57
From the above we can see that the domain of sin
–1
x + sinx is the intersection of the domains of sin
–1
x and
sin x.
So domain of sin
–1
x + sinx is [–1, 1].
3 C. Question
Find the domain of each of the following functions:
f(x) = 
Answer
Domain of  lies in the interval [–1, 1].
Therefore, Domain of  lies in the interval [–1, 1].
–1  1
0  1
1  2
 and 
Domain of  is [– [1, .
3 D. Question
Find the domain of each of the following functions:
f(x) = sin
–1
x + sin
–1
2x
Answer
Domain of  lies in the interval [–1, 1].
–1 1
Therefore, the domain of  lies in the interval 
–1  1
The domain of sin
–1
x + sin
–1
2x is the intersection of the domains of sin
–1
x and sin
–1
2x.
So, Domain of sin
–1
x + sin
–1
2x is .
4. Question
If sin
–1
x + sin
–1
y + sin
–1
z + sin
–1
t = 2p, then find the value of
x
2
 + y
2
 + z
2
 + t
2
.
Answer
Range of sin
–1
x is .
Give that sin
–1
x + sin
–1
y + sin
–1
z + sin
–1
t = 2p
Each of sin
–1
x, sin
–1
y, sin
–1
z, sin
–1
t takes value of .
So,
x = 1, y = 1, z = 1 and t = 1.
Hence,
= x
2
 + y
2
 + z
2
 + t
2
= 1 + 1 + 1 + 1
= 4
5. Question
If (sin
–1
x)
2
 + (sin
–1
y)
2
 + (sin
–1
z)
2
 = 3/4 p
2
. Find x
2
 + y
2
 + z
2
.
Answer
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