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

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