Question

Class 12th, Inverse Trigonometric Function

Simplify the following expressions

Answer 1

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How to simplify expressions including inverse trigonometric functions for grade 12 math. Detailed solutions are also included.

Simplify the expressions: 
a) sin(arcsin(x)) and arcsin(sin(x)) 
b) cos(arccos(x)) and arccos(cos(x)) 
c) tan(arctan(x)) and arctan(tan(x)) 
Solution 
a) sin and arcsin are inverse of each other and therefore the properties of inverse functions may be used to write 
sin(arcsin(x)) = x , for -1 ≤ x ≤ 1 
arcsin(sin(x)) = x , for x ∈ [-π/2 , π/2] 
NOTE: If x in arcsin(sin(x)) is not in the interval [-π/2 , π/2], find θ in the interval [-π/2 , π/2] such that sin(x) = sin(θ) and then simplify arcsin(sin(x)) = θ 
b) cos and arccos are inverse of each other and therefore the properties of inverse functions may be used to write 
cos(arccos(x)) = x , for -1 ≤ x ≤ 1 
arccos(cos(x)) = x , for for x ∈ [0 , π] 
NOTE: If x in arccos(cos(x)) is not in the interval [0/2 , π], find θ in the interval [0 , π] such that cos(x) = cos(θ) and then simplify arccos(cos(x)) = θ 
c) tan and arctan are inverse of each other and therefore the properties of inverse functions may be used to write 
tan(arctan(x)) = x 
arctan(tan(x)) = x , for x ∈ (-π/2 , π/2) 
NOTE: If x in arctan(tan(x)) is not in the interval (-π/2 , π/2), find θ in the interval (-π/2 , π/2) such that tan(x) = tan(θ) and then simplify arctan(tan(x)) = θ 

Express the following as algebraic expressions: 
sin(arccos(x)) and tan(arccos(x)) 
Solution 
Let A = arccos(x). Hence 
cos(A) = cos(arccos(x)) = x 
Use right triangle with angle A such that cos(A) = x (or x / 1), find second leg and calculate sin(A) and tan(A)

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sin(arccos(x)) = sin(A) = √(1 - x2) / 1 = √(1 - x2)     for x ∈ [-1 , 1] 
tan(arccos(x)) = tan(A) = √(1 - x2) / x     for x ∈ [-1 , 0) ∪ (0 , 1]


.got it.