Formulas: Inverse Trigonometric Functions

# Formulas: Inverse Trigonometric Functions | Mathematics (Maths) Class 11 - Commerce PDF Download

General Definition(s): 1. sin−1 x , cos−1 x , tan−1 x etc. denote angles or real numbers whose sine is x , whose cosine is x and whose tangent is x, provided that the answers given are numerically smallest available . These are also written as are sinx , are cosx etc . If there are two angles one positive & the other negative having same numerical value, then positive angle should be taken.

2. Principal Values And Domains Of Inverse Functions :
(i) y = sin−1 x where −1 ≤ x ≤ 1 ;  and sin y = x.
(ii) y = cos−1 x where −1 ≤ x ≤ 1 ; 0 ≤ y ≤ π and cos y = x.
(iii) y = tan−1 x where x ∈ R ;  and tan y = x.
(iv) y = cosec−1 x where x ≤ −1 or x ≥ 1;  y ≠ 0 and cosec y = x
(v) y = sec−1 x where x ≤ −1  or  x ≥ 1 ; 0 ≤ y ≤ π  ;  y ≠ π/2 and sec y = x.
(vi) y = cot−1 x where x ∈ R, 0 < y < π and cot y = x.
NOTE THAT : (a) 1st quadrant is common to all the inverse functions.
(b) 3rd quadrant is not used in inverse functions.
(c) 4th quadrant is used in the CLOCKWISE DIRECTION i.e. .

3. PROPERTIES OF INVERSE CIRCULAR FUNCTIONS :
P−1
(i) sin (sin−1 x) = x, −1 ≤ x ≤ 1
(ii) cos (cos−1 x) = x, −1 ≤ x ≤ 1
(iii) tan (tan−1 x) = x, x ∈ R
(iv) sin−1 (sin x) = x,
(v) cos−1 (cos x) = x; 0 ≤ x ≤ π
(vi) tan−1 (tan x) = x ;

P−2
(i) cosec−1 x = sin−1(1/x) ; x ≤ −1, x ≥ 1
(ii) sec−1 x = cos−1 (1/x) ; x ≤ −1, x ≥ 1
(iii)  cot−1 x = tan−1 (1/x) ; x > 0 = π + tan−1(1/x) ; x < 0

P−3
(i) sin−1 (−x) = − sin−1x, −1 ≤ x ≤ 1
(ii) tan−1 (−x) = − tan−1 x, x ∈ R
(iii) cos−1 (−x) = π − cos−1 x, −1 ≤ x ≤ 1
(iv) cot−1 (−x) = π − cot−1 x, x ∈ R

P−4
(i) sin−1 x + cos−1 x = (π/2)  −1 ≤ x ≤ 1
(ii) tan−1 x + cot−1 x = (π/2)  x ∈ R
(iii) cosec−1 x + sec−1 x = (π/2) |x| ≥ 1

P−5
tan−1 x + tan−1 y = tan−1  where x > 0, y > 0 & xy < 1
= π + tan−1 where x > 0,  y > 0 & xy > 1
tan−1 x − tan−1y = tan−1 where  x > 0, y > 0

P−6
(i) sin−1 x + sin−1 y = sin−1  where x ≥ 0, y ≥ 0 & (x2 + y2) ≤ 1
Note that : x2 + y2 ≤ 1 ⇒ 0 ≤ sin−1 x + sin−1 y ≤ π/2
(ii) sin−1 x + sin−1 y = π − sin−1 where x ≥ 0, y ≥ 0 & x2 + y2 > 1
Note that : x2 + y2 >1 ⇒ π/2 < sin−1 x + sin−1 y < π
(iii) sin–1x – sin–1y = 1  where x > 0, y > 0
(iv) cos−1 x + cos−1 y = cos−1 where x ≥ 0, y ≥ 0

P-7
If tan−1 x + tan−1 y + tan−1 z = tan−1 if, x > 0, y > 0, z > 0 & xy + yz + zx < 1
Note : (i) If tan−1 x + tan−1 y + tan−1 z =  π then x + y + z = xyz
(ii) If tan−1 x + tan−1 y + tan−1 z = π/2 then xy + yz + zx = 1

P-8
2 tan−1 x =  Note very carefully that :

REMEMBER  THAT :
(i) sin−1 x + sin−1 y + sin−1 z =  ⇒ x = y = z = 1
(ii) cos−1 x + cos−1 y + cos−1 z = 3π ⇒ x = y = z = −1
(iii) tan−1 1 + tan−1 2 + tan−1 3 = π and tan−1 1 + tan−1

INVERSE TRIGONOMETRIC FUNCTIONS (SOME USEFUL GRAPHS)

1. y = sin−1 x, |x| ≤ 1, y ∈

2. y = cos−1 x, |x| ≤ 1, y ∈ [0, π]

3. y = tan−1 x , x ∈ R,

4. y = cot−1 x, x ∈ R, y ∈ (0, π)

5. y = sec−1 x, |x| ≥ 1,

6. y = cosec−1 x, |x| ≥ 1,

7. (a) y = sin−1 (sin x), x ∈ R,
Periodic with period 2π aperiodic

(b) y = sin (sin−1 x), = x  x ∈ [−1, 1],  y ∈ [−1, 1] , y is

8. (a) y = cos−1(cos x),  x ∈ R, y ∈ [0, π], = x periodic with period 2π

(b) y = cos (cos−1 x), = x
x ∈ [−1, 1], y ∈ [−1, 1], y is aperiodic

9. (a) y = tan (tan−1 x),  x ∈ R,  y ∈ R,  y is aperiodic = x

(b) y = tan−1 (tan x), = x

10. (a) y = cot−1 (cot x), =  x
x ∈ R − {n π}, y ∈ (0, π), periodic with π

(b) y = cot (cot−1 x), = x
x ∈ R,  y ∈ R, y is aperiodic

11. (a) y = cosec−1 (cosec x), = x

(b) y = cosec (cosec−1 x), = x
|x| ≥ 1, |y| ≥ 1, y is aperiodic

12. (a) y = sec−1 (sec x), = x
y is periodic with period 2π ;

(b) y = sec (sec−1 x), = x
|x| ≥ 1 ; |y| ≥ 1], y is aperiodic

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## FAQs on Formulas: Inverse Trigonometric Functions - Mathematics (Maths) Class 11 - Commerce

 1. What are inverse trigonometric functions?
Ans. Inverse trigonometric functions, also known as arc trigonometric functions, are functions that provide the angle or angles in a right triangle when the ratios of its sides are given. They are the inverse operations of trigonometric functions such as sine, cosine, and tangent.
 2. How do inverse trigonometric functions relate to trigonometric functions?
Ans. Inverse trigonometric functions are used to find the angles in a right triangle when the ratios of its sides are known. They provide a way to "undo" the trigonometric functions and find the angle that corresponds to a given ratio. For example, if we know the sine of an angle, we can use the inverse sine function to find the angle itself.
 3. What are the common inverse trigonometric functions?
Ans. The common inverse trigonometric functions are: - Inverse sine (arcsin or sin^-1): Gives the angle whose sine is a given ratio. - Inverse cosine (arccos or cos^-1): Gives the angle whose cosine is a given ratio. - Inverse tangent (arctan or tan^-1): Gives the angle whose tangent is a given ratio.
 4. How are inverse trigonometric functions represented?
Ans. Inverse trigonometric functions are typically represented using the prefix "arc" or with the exponent of "-1" (e.g., arcsin or sin^-1). They can also be represented using abbreviations such as asin, acos, and atan. The notation used may vary depending on the context or mathematical convention.
 5. What is the range of inverse trigonometric functions?
Ans. The range of inverse trigonometric functions depends on the specific function. - Inverse sine (arcsin or sin^-1) has a range of [-π/2, π/2], which means it gives angles between -90 degrees and 90 degrees. - Inverse cosine (arccos or cos^-1) has a range of [0, π], which means it gives angles between 0 degrees and 180 degrees. - Inverse tangent (arctan or tan^-1) has a range of [-π/2, π/2], which means it gives angles between -90 degrees and 90 degrees. These ranges ensure that the inverse trigonometric functions provide unique values for a given ratio and avoid ambiguity.

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