How to simplify expressions including inverse trigonometric functions for grade 12 maths. Questions with detailed solutions are also included.

a) sin(arcsin(x)) and arcsin(sin(x))

b) cos(arccos(x)) and arccos(cos(x))

c) tan(arctan(x)) and arctan(tan(x))

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 ? [-?/2 , ?/2]

NOTE: If x in arcsin(sin(x)) is not in the interval [-?/2 , ?/2], find ? in the interval [-?/2 , ?/2] such that sin(x) = sin(?) and then simplify arcsin(sin(x)) = ?

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 , ?]

NOTE: If x in arccos(cos(x)) is not in the interval [0/2 , ?], find ? in the interval [0 , ?] such that cos(x) = cos(?) and then simplify arccos(cos(x)) = ?

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 ? (-?/2 , ?/2)

NOTE: If x in arctan(tan(x)) is not in the interval (-?/2 , ?/2), find ? in the interval (-?/2 , ?/2) such that tan(x) = tan(?) and then simplify arctan(tan(x)) = ?

sin(arccos(x)) and tan(arccos(x))

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)

.

sin(arccos(x)) = sin(A) = √(1 - x

tan(arccos(x)) = tan(A) = √(1 - x

cos(arcsin(x)) and tan(arcsin(x))

Let A = arcsin(x). Hence

sin(A) = sin(arcsin(x)) = x

Use right triangle with angle A such that sin(A) = x (or x / 1), find second leg and calculate cos(A) and tan(A)

.

cos(arcsin(x)) = cos(A) = √(1 - x

tan(arcsin(x)) = tan(A) = x / √(1 - x

sin(arctan(x)) and cos(arctan(x))

Let A = arctan(x). Hence

tan(A) = tan(arctan(x)) = x

Use right triangle with angle A such that tan(A) = x (or x / 1), find hypotenuse and calculate sin(A) and cos(A)

.

sin(arctan(x)) = sin(A) = x / √(1 + x

cos(arctan(x)) = cos(A) = 1 / √(1 + x

a) arccos(0) , arcsin(-1) , arctan(-1)

b) sin(arcsin(-1/2)) , arccos(cos(?/2)) , arccos(cos(-?/2))

c) cos(arcsin(-1/2)) , arcsin(sin(?/3)) , arcsin(tan(3?/4))

d) arccos(tan(7?/4)) , arcsin(sin(13?/3)) , arctan(tan(-17?/4)) , arcsin(sin(9?/5))

a) Use definition.

arccos(0) = ?/2 because cos(?/2) = 0 and ?/2 is within range of arccos which is [0 , ?]

arcsin(-1) = -?/2 because sin(-?/2) = -1 and -?/2 is within range of arcsin which is [-?/2 , ?/2]

arctan(-1) = -?/4 because tan(-?/4) = -1 and -?/4 is within range of arctan which is (-?/2 , ?/2)

b) Simplify inner functions then the outer functions using definitions.

sin(arcsin(-1/2)) = sin(-?/6) = -1/2

arccos(cos(?/2)) = arccos(0) = ?/2

arccos(cos(-?/2)) = arccos(0) = ?/2

c) Simplify inner functions then the outer functions using definitions.

cos(arcsin(-1/2)) = cos(-?/6) = √3/2

arcsin(sin(?/3)) = arcsin(√3/2) = ?/3

arcsin(tan(3?/4)) = arcsin(-1) = -?/2

d) Simplify inner functions then the outer functions using definitions.

arccos(tan(7?/4)) = arccos(-1) = ?

arcsin(sin(13?/3)) = arcsin(sin(4? + ?/3)) = arcsin(sin(?/3)) = ?/3

arctan(tan(- 17?/4)) = arctan(tan(- 4?-?/4)) = arctan(tan(- ?/4)) = - ?/4

arcsin(sin(9?/5)) = arcsin(sin(2? - ?/5)) = arcsin(sin(- ?/5)) = - ?/5

Use the indentity sin(A + B) = sin(A)cos(B) + cos(A)sin(B) to expand the given expression.

sin(A + B) = sin(arcsin(2/3))cos(arccos(-1/2)) + cos(arcsin(2/3))sin(arccos(-1/2))

Use the above indentities to simplify each term in the above expression.

sin(arcsin(2/3)) = 2/3 (we have used sin(arcsin(x)) = x))

cos(arccos(-1/2)) = -1/2 (we have used cos(arccos(x)) = x))

cos(arcsin(2/3)) = √(1 - (2/3)

sin(arccos(-1/2)) = √(1 - (- 1/2)

sin(A + B) = (2/3)(-1/2)+(√5/3)(√3/2) = -1/3 + √(15)/6

Let A = arcsin(x). Hence Y may be written as

Y = sin (2 A)

Use the identity sin(2 A) = 2 sin(A) cos(A) to rewrite Y as folllows:

Y = 2 sin (A) cos(A) = 2 sin(arcsin(x)) cos(arcsin(x))

Use the identities sin(arcsin(x)) = x and cos(arcsin(x)) = √(1-x

Y = 2 x √(1 - x

Let A = arctan(3/4). Hence Y may be written as

Y = sin(2 A) = 2 sin(A) cos(A)

sin(A) = sin(arctan(3/4)) = (3/4) / √(1 + (3/4)

cos(A) = cos(arctan(3/4)) = 1 / √(1 + (3/4)

Y = 2 (3 / 5)(4 / 5) = 24 / 25

Solve Inverse Trigonometric Functions Questions

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