Class 12 Maths

# Inverse Trignometric Functions

## NCERT Solution

### Miscellaneous Exercise 2 Part 1

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

Question 1: text(cos)^(-1)(text(cos)(13π)/6)

Solution: Given, text(cos)^(-1)(text(cos)(13π)/6)

=text(cos)^(-1)[text(cos)(2π-(π)/(6))

=text(cos)^(-1)(text(cos)(π)/6)

Since, (π)/6∈[0,π]

Hence, text(cos)^(-1)(text(cos)(13π)/6)=(π)/6

Question 2: text(tan)^(-1)(text(tan)(7π)/6)

Solution: Since (7π)/6∈(-(π)/2, (π)/2)

Hence, text(tan)^(-1)(text(tan)(7π)/6)=text(tan)^(-1)[text(tan)(π+(π)/6)]

=text(tan)^(-1)(text(tan)(π)/6)

Now, as (π)/6∈(-(π)/2, (π)/2)

Hence, text(tan)^(-1)(text(tan)(7π)/6)=(π)/6

Question 3: Prove that 2text(sin)^(-1)3/5=text(tan)^(-1)(24)/7

Solution: LHS =2text(sin)^(-1)3/5

=text(sin)^(-1)(2xx3/5sqrt(1-(3/5)^2)

Because 2text(sin)^(-1)x=text(sin)^(-1)(2x\sqrt(1-x^2))

=text(sin)^(-1)(6/5sqrt(1-9/(25)))

=text(sin)^(-1)(6/5sqrt((16)/(25)))

=text(sin)^(-1)(6/5xx4/5)=text(sin)^(-1)(24)/(25)

Since, text(sin)^(-1)x=text(tan)^(-1)x/(sqrt(1-x^2))

Hence, text(sin)^(-1)(24)/(25)=text(tan)^(-1)((24)/(25))/(sqrt(1-((24)/(25))^2))

=text(tan)^(-1)(((24)/(25))/(sqrt(1-(576)/(625))))

=text(tan)^(-1)(((24)/(25))/(sqrt(625-576)/(625)))

=text(tan)^(-1)(((24)/(25))/(sqrt((49)/(625))))

=text(tan)^(-1)((24)/(25)xx(25)/7)

=text(tan)^(-1)(24)/7=RHS proved

Question 4: text(sin)^(-1)8/(17)+text(sin)^(-1)3/5=text(tan)^(-1)(77)/(36)

Solution: LHS = text(sin)^(-1)8/(17)+text(sin)^(-1)3/5

=text(sin)^(-1)[8/(17)sqrt(1-(3/5)^2)+3/5sqrt(1-(8/(17))^2)]

=text(sin)^(-1)[8/(17)sqrt((25-9)/(25))+3/5sqrt((289-64)/(289))

=text(sin)^(-1)[8/(17)sqrt((16)/(25))+3/5sqrt((225)/(289))]

text(sin)^(-1)[8/(17)xx4/5+3/5xx(15)/(17)]

=text(sin)^(-1)[(32)/(85)+(45)/(85)=text(sin)^(-1)(77)/(85)

=text(tan)^(-1)[((77)/(85))/(sqrt(1-((77)/(85))^2)]

Because text(sin)^(-1)x=text(tan)^(-1)x/(sqrt(1-x^2)

=text(tan)^(-1)[((77)/(85))/(sqrt((7225-5925)/(7225)))]

=text(tan)^(-1)[((77)/(85))/(sqrt((126)/(7225)))]

=text(tan)^(-1)[((77)/(85))/((36)/(85))]

=text(tan)^(-1)(77)/(36)= RHS proved

Question 5: text(cos)^(-1)4/5+text(cos)^(-1)(12)/(13)=text(cos)^(-1)(33)/(65)

Solution: LHS =text(cos)^(-1)4/5+text(cos)^(-1)(12)/(13)

Since text(cos)^(-1)x+text(cos0^(-1)y=text(cos)^(-1)[xy-sqrt(1-x^2)sqrt(1-y^2)]

Hence, LHS can be written as follows:

text(cos)^(-1)[4/5xx(12)/(13)-sqrt(1-(4/5)^2)sqrt(1-((12)/(13))^2)]

=text(cos)^(-1)[(48)/(65)-sqrt(1-(16)/(25))sqrt(1-(144)/(169))]

=text(cos)^(-1)(48)/(65)-3/5xx5/(13)]

=text(cos)^(-1)[(48)/(65)-3/(13)]

=text(cos)^(-1)[(48-15)/(65)]=text(cos)^(-1)(33)/(65)= RHS proved