EVALUATING COMPOSITION OF TRIG AND INVERSE FUNCTIONS
To find the exact value of inverse trigonometric function, we have to ask the question that for which angle we will get the given value.
Find the exact value without a calculator.
Problem 1 :
cossin-112
Solution:
=cossin-112The principal solution will lie in -𝜋2,𝜋2.=cos𝜋6=32
Problem 2 :
sincos-122
Solution:
=sincos-122The principal solution will lie in [0, π].=sin𝜋4=12=22
Problem 3 :
sin-1cos𝜋3
Solution:
=sin-1cos𝜋3=sin-112The principal solution will lie in -𝜋2,𝜋2.=𝜋6
Problem 4 :
cos-1sin𝜋6
Solution:
=cos-1sin𝜋6=cos-112The principal solution will lie in [0, π].=𝜋3
Problem 5 :
sin-1sin7𝜋4
Solution:
=sin-1sin7𝜋4=sin-1sin2𝜋-𝜋42𝜋-𝜋4 lies in the fourth quadrant.=sin-1sin7𝜋4=sin-1-22=-𝜋4
Problem 6 :
arccossin𝜋3
Solution:
=arccossin𝜋3=cos-132The principal solution will lie in [0, π].=𝜋6
Problem 7 :
sintan-13
Solution:
=sintan-13=sin(60°)=32
Problem 8 :
costan-1(-1)
Solution:
=costan-1(-1)The principal solution will lie in -𝜋2,𝜋2.=cos-𝜋4=12=22
Problem 9 :
tan-1(cos(𝜋))
Solution:
=tan-1(cos(𝜋))=tan-1(-1)The principal solution will lie in -𝜋2,𝜋2.=-𝜋4