Class 12 Maths

# Inverse Trignometric Functions

## NCERT Solution

### Exercise 2.1 Part 1

##### Find the principal values of the following:

Question 1: text(sin)^(-1)(-1/2)

Solution: Let text(sin)^(-1)(-1/2)=θ

Therefore, text(sin)θ=-1/2

Or, text(sin)θ=-text(sin)30°

Or, text(sin)θ=-text(sin)((π)/(6))

Or, text(sin)θ=text(sin)((-π)/(6))

Since principal value of text(sin)^(-1)x lies between -(π)/(2) and (π)/(2)

Hence, principal value of text(sin)^(-1)(-1/2) is -(π)/(6)

Question 2: text(cos)^(-1)((sqrt3)/(2))

Solution: Let text(cos)^(-1)((sqrt3)/(2))=θ

Hence, text(cos)θ=(sqrt3)/2

Or, text(cos)θ=text(cos)((π)/(6))

Since principal value of text(cos)^(-1) lies between 0 and π. Hence, principal value of text(cos)^(-1)((sqrt3)/(2)) is (π)/6

Question 3: text(cosec)^(-1)(2)

Solution: Let text(cosec)^(-1)(2)=θ

So, text(cosec)θ=2

Or, text(cosec)θ=text(cosec)((π)/(6))

Since principal value of text(cosec)^(-1)x is [(-π)/(2), (π)/(2)]-(0)

Hence, principal value of text(cosec)^(-1)(2) is (π)/6

Question 4: text(tan)^(-1)(-sqrt3)

Solution: Let text(tan)^(-1)(-sqrt3)=θ

So, text(tan)θ=-sqrt3

Or, text(tan)θ=-text(tan)((π)/(3))

Or, text(tan)θ=text(tan)((-π)/(3))

Since value of text(tan)^(-1)x lies between (-π)/2 and (π)/2

Hence, principal value of text(tan)^(-1)(-sqrt3) is (-π)/3

Question 5: text(cos)^(-1)(-1/2)

Solution: Let text(cos)^(-1)(-1/2)=θ

So, text(cos)θ=-1/2

Or, text(cos)θ=-text(cos)((π)/(3))

Or, text(cos)θ=text(cos)(π-(π)/(3))

Or, text(cos)θ=text(cos)((2π)/(3))

We know that the range of the principal value of text(cos)^(-1) is [0, x] and text(cos)((2π)/(3))=-1/2

Hence, principal value of text(cos)^(-1)(-1/2) is (2π)/3

Question 6: text(tan)^(-1)(-1)

Solution: Let text(tan)^(-1)(-1)=θ

So, text(tan)θ=-1

Or, text(tan)θ=-text(tan)((π)/(4))

Or, text(tan)θ=text(tan)(-(π)/(4))

Since value of text(tan)^(-1)x lies between -(π)/2 and (π)/2

Hence, principal value of text(tan)^(-1)(-1) is (-π)/4

Question 7: text(sec)^(-1)((2)/(sqrt3))

Solution: Let text(sec)^(-1)((2)/(sqrt3))=θ

So, text(sec)θ=2/(sqrt3)

Or, text(sec)θ=text(sec)(π)/6

Hence, principal value of text(sec)^(-1)((2)/(sqrt3)) is (π)/6

Question 8: text(cot)^(-1)(sqrt3)

Solution: Let text(cot)^(-1)(sqrt3)=θ

So, text(cot)θ=sqrt3

Or, text(cot)θ=text(cot)(π)/6

Thus, principal value of text(cot)^(-1)(sqrt3) is (π)/6`

Exercise 2.1 Part 1

Exercise 2.1 Part 2

Exercise 2.2 Part 1

Exercise 2.2 Part 2

Exercise 2.2 Part 3

Exercise 2.2 Part 4

Miscellaneous Exercise