Inverse trigonometric functions are the inverse functions of basic trigonometric functions. In mathematics, inverse trigonometric functions are also known as arcus functions or anti-trigonometric functions.
- The inverse trigonometric functions are the inverse functions of basic trigonometric functions, i.e., sine, cosine, tangent, cosecant, secant, and cotangent.
- It is used to find the angles with any trigonometric ratio.
- Inverse trigonometric functions are generally used in fields like geometry, engineering, etc.
The table below discusses the inverse trigonometric function formulas for negative functions.
| Inverse Trig Functions | Formulas |
|---|
| Arcsine | sin-1(-x) = -sin-1(x), x ∈ [-1, 1] |
| Arccosine | cos-1(-x) = π -cos-1(x), x ∈ [-1, 1] |
| Arctangent | tan-1(-x) = -tan-1(x), x ∈ R |
| Arccotangent | cot-1(-x) = π – cot-1(x), x ∈ R |
| Arcsecant | sec-1(-x) = π -sec-1(x), |x| ≥ 1 |
| Arccosecant | cosec-1(-x) = -cosec-1(x), |x| ≥ 1 |
The inverse trigonometric function for reciprocal values of x converts the given inverse trigonometric function into its reciprocal function. This follows from the trigonometric functions where sin and cosecant are reciprocal to each other, tangent and cotangent are reciprocal to each other, and cos and secant are reciprocal to each other.
The inverse trigonometric formula of inverse sine, inverse cosine, and inverse tangent are expressed as,
- sin-1x = cosec-11/x, x ∈ R - (-1, 1)
- cos-1x = sec-11/x, x ∈ R - (-1, 1)
- tan-1x = cot-11/x, x > 0
Complementary functions are the functions whose addition results in right angles. Thus, the sum of the complementary inverse trigonometric functions also results in a right angle.
The complementary function pairs are, sine-cosine, tangent-cotangent, and secant-cosecant i.e. for a similar function the sum of these functions results in the right angle. Various inverse trigonometric complementary functions are,
- sin-1x + cos-1x = π/2, x ∈ [-1,1]
- tan-1x + cot-1x = π/2, x ∈ R
- sec-1x + cosec-1x = π/2, x ∈ R - [-1,1]
Sum and difference of two inverse trigonometric functions can be combined to form a single inverse function, as per the below set of formulas. The sum and the difference of the inverse trigonometric functions have been derived from the trigonometric function formulas of sin (A + B), cos (A + B), and tan (A + B). These inverse trigonometric function formulas can be used to further derive the double and triple function formulas.
- sin-1x + sin-1y = sin-1(x.√(1 - y2) + y√(1 - x2))
- sin-1x - sin-1y = sin-1(x.√(1 - y2) - y√(1 - x2))
- cos-1x + cos-1y = cos-1(xy - √(1 - x2).√(1 - y2))
- cos-1x - cos-1y = cos-1(xy + √(1 - x2).√(1 - y2))
- tan-1x + tan-1y = tan-1(x + y)/(1 - xy), if xy < 1
- tan-1x + tan-1y = tan-1(x - y)/(1 + xy), if xy > - 1
Double inverse trigonometric function formulas are the formulas that give the values of the double angle in the inverse trigonometric function. Some important double inverse trigonometric function formulas are,
- 2sin-1x = sin-1(2x.√(1 - x2))
- 2cos-1x = cos-1(2x2 - 1)
- 2tan-1x = tan-1(2x/1 - x2)
These formulas are derived using the basic double-angle formulas of the trigonometric function.
Triple inverse trigonometric function formulas are the formulas that give the values of the triple angle in the inverse trigonometric function. Some important triple inverse trigonometric function formulas are,
- 3sin-1x = sin-1(3x - 4x3)
- 3cos-1x = cos-1(4x3 - 3x)
- 3tan-1x = tan-1(3x - x3/1 - 3x2)
These formulas are derived using the basic triple-angle formulas of the trigonometric function.
Inverse Trigonometric Functions Domain and Range
The domain and the range of the inverse trigonometric function are added in the table below.
| Function | Domain | Range |
|---|
| y = sin-1 x | [-1, 1] | [-π/2, π/2] |
| y = cos-1 x | [-1, 1] | [0 , π] |
| y = cosec-1 x | R – (-1,1 ) | [-π/2, π/2] – {0} |
| y = sec-1 x | R – (-1, 1) | [0 , π] – {π/2} |
| y = tan-1 x | R | (-π/2, π/2) |
| y = cot-1 x | R | (0 , π) |
Inverse Trigonometric Function Types
There are a total of six Inverse Trigonometric Functions that are,
Arcsine Function
Arcsine function is an inverse of the sine function denoted by sin-1x. It returns the angle whose sine corresponds to the provided number.
sin θ = (Opposite/Hypotenuse)
=> sin-1 (Opposite/Hypotenuse) = θ
Example: sin-1(1/2) = π/6
Theorem of sin inverse is: d/dx sin-1x = 1/√(1 - x2)
Proof:
sin(θ) = x
Now,
f(x) = sin-1x ..(eq1)
substitute value of sin in eq(1)
f(sin(θ)) = θ
f'(sin(θ))cos(θ) = 1 .. differentiating equation
we know that,
sin2θ + cos2θ= 1
So, cos = √(1 - x2)
f'(x) = 1/√(1 - x2)
Now,
d/dx sin-1x = 1/√(1 - x2)
Hence Proved.
Arccosine Function
Arccosine function is an inverse of the sine function denoted by cos-1. It returns the angle whose cosine corresponds to the provided number.
cos θ = (Hypotenuse/Adjacent)
=> cos-1 (Hypotenuse/Adjacent) = θ
Example: cos-1(1/2) = π/3
Theorem of cos inverse is: d/dx cos-1(x) = -1/√(1 - x2)
Proof:
cos(θ) = x
θ = arccos(x)
dx = dcos(θ) = −sin(θ)dθ .. differentiate the equation
Now,
we know that,
sin2 + cos2 = 1
So, cos = √(1 - x2)
−sin(θ) = −sin(arccos(x)) = -√(1 - x2)
dθ/dx = −1/sin(θ) = -1/√(1 - x2)
So,
dθ/dx cos-1(x) = -1/√(1 - x2)
Hence Proved.
Arctangent Function
Arctangent function is an inverse of the tangent function denoted by tan-1. It returns the angle whose tangent corresponds to the provided number.
tan θ = (Opposite/Adjacent)
=> tan-1 (Opposite/Adjacent) = θ
Example: tan-1(1) = π/4
Theorem of tan inverse is: d/dx tan-1(x) = 1/(1 + x2)
Proof:
tan(θ) = x
θ = arctan(x)
We know that,
tan2θ + 1 = sec2θ
dx/dθ = sec2y .. differentiating tan function
dx/dθ = 1+x2
Therefore,
dθ/dx = 1/(1 + x2)
Hence Proved.
Arccotangent Function
Arccotangent function is an inverse of the tangent function denoted by cot-1. It returns the angle whose tangent corresponds to the provided number.
cot θ = (Adjacent/Opposite)
=> cot-1 (Adjacent/opposite) = θ
Example: cot-1(1) = π/4
Arcsecant Function
Arcsecant function is an inverse of the secant function denoted by sec-1. It returns the angle whose secant corresponds to the provided number.
sec θ = (Base/Hypotenuse)
=> sec-1 (Base/Hypotenuse) = θ
Example: sec-1(2) = π/3
Arccosecant Function
Arccosecant function is an inverse of the cosecant function denoted by cosec-1. It returns the angle whose cosecant corresponds to the provided number.
cosec θ = (Adjacent/Hypotenuse)
=> Cossec-1 (Adjacent/Hypotenuse) = θ
Example: Cosec-1(2) = π/6
Restricting Domains of Functions to Make Them Invertible
A real function in the range ƒ: R ⇒ [-1, 1] defined by ƒ(x) = sin(x) is not a bijection since different images have the same image such as ƒ(0) = 0, ƒ(2π) = 0,ƒ(π) = 0, so, ƒ is not one-one. Since ƒ is not a bijection (because it is not one-one) therefore inverse does not exist. To make a function bijective we can restrict the domain of the function to [−π/2, π/2] or [−π/2, 3π/2] or [−3π/2, 5π/2] after restriction of domain ƒ(x) = sin(x) is a bijection, therefore ƒ is invertible. i.e. to make sin(x) we can restrict it to the domain [−π/2, π/2] or [−π/2, 3π/2] or [−3π/2, 5π/2] or...…. but [−π/2, π/2] is the Principal solution of sinθ, hence to make sinθ invertible. Naturally, the domain [−π/2, π/2] should be considered if no other domain is mentioned.
- ƒ:[−π/2, π/2] ⇒ [-1 , 1] is defined as ƒ(x) = sin(x) and is a bijection, hence inverse exists. The inverse of sin-1 is also called arcsine and inverse functions are also called arc functions.
- ƒ:[−π/2 , π/2] ⇒ [−1 , 1] is defined as sinθ = x ⇔ sin-1(x) = θ , θ belongs to [−π/2 , π/2] and x belongs to [−1 , 1].
Similarly, we restrict the domains of cos, tan, cot, sec, and cosec so that they are invertible.
Derivatives of Inverse Trigonometric Functions
We can find the differentiation of inverse trigonometric functions using differentiation formulas. The image added below shows the same
The following table gives the result of the differentiation of inverse trig functions.
Inverse Trigonometric Functions | dy/dx |
|---|
| y = sin 1(x), x ≠ -1, +1 | 1/√(1-x2) |
| y = cos 1(x), x ≠ -1, +1 | -1/√(1-x2) |
| y = tan-1(x), x ≠ -i, +i | 1/(1+x2) |
| y = cot-1(x), x ≠ -i, +i | -1/(1+x2) |
| y = sec 1(x), |x| > 1 | 1/[|x|√(x2-1)] |
| y = cosec 1(x), |x| > 1 | -1/[|x|√(x2-1)] |
Integrals of Inverse Trigonometric Functions
Here are the integral formulas of inverse trigonometric functions. To see how to derive each one of them,
| Inverse Trigonometric Function | Integral |
|---|
| ∫ sin-1x dx | x sin-1x + √ (1 - x²) + C |
| ∫ cos-1x dx | x cos-1x - √ (1 - x²) + C |
| ∫ tan-1x dx | x tan-1x - (1/2) ln |1 + x²| + C |
| ∫ csc-1x dx | x csc-1x + ln |x + √ (x² - 1)| + C |
| ∫ sec-1x dx | x sec-1x - ln |x + √ (x² - 1) | + C |
| ∫ cot-1x dx | x cot-1x + (1/2) ln |1 + x²| + C |
Related Articles:
Examples on Inverse Trigonometric Functions
Example 1: Find x, if sin(x) = 1/2
Solution:
Given, sin x = 1/2
Using inverse trigonometric function formulas,
x = sin-1(1/2)
Using Trigonometric Table
x = sin-1[sin(π/6)]
x = π/6
Thus, the required value of x is π/6
Example 2: Find the value of cos-1(1/2) - sec-1(-2).
Solution:
cos-1 (1/2) - sec-1(-2)
= π/3 - (π - sec-12)
Using Trigonometric Table
= π/3 - (π - π/3)
= π/3 - π + π/3
= π/3 + π/3 - π
= 2π/3 - π
= -π/3
Thus, the required value of the given expression is -π/3
Example 3: Find cos [cos-1 (11/15)]
Solution:
= cos [cos-1 (11/15)]
We know that, cos [cos-1 (x)] = x
= 11/15
Example 4: Find the value of cot-1(1) + sin-1(-1/2) + sin-1(1/2).
Solution:
Using inverse trigonometric functions formulas,
= cot-1(1) + sin-1(-1/2) + sin-1(1/2)
= cot-1(1) - sin-1(1/2) + sin-1(1/2)
= cot-1(1)
= π/4
Thus, the required solution is π/4
Example 5: Simplify sin(tan-1x)
Solution:
Taking sin2 (θ) + cos2 (θ) = 1
dividing both sides by sin2(θ)
1 + cot2 (θ) = cosec2 (θ)
1 + 1/tan2 (θ) = 1/sin2 (θ)
(tan2 (θ) + 1)/tan2 (θ) = 1/sin2 (θ)
Taking the inverse
sin2 (θ) = tan2 (θ)/(tan2 (θ) + 1)...(1)
taking θ = tan-1(x) In eq (1)
sin2 (tan-1(x)) = tan2 (tan-1(x))/(tan2 (tan-1(x)) + 1)
We know that, tan (tan-1(x)) = x
sin2 (tan-1(x)) = x2 / x2 + 1
Taking square root on both sides,
sin (tan-1(x)) = x / √(x2 + 1)
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