Basics Concepts: Inverse Trigonometric Functions

## What is an Inverse Function?

• A function accepts values, performs particular operations on these values, and generates an output. The inverse function agrees with the resultant, operates, and reaches back to the original function.
• If y=f(x) and x=g(y) are two functions such that f (g(y)) = y and g (f(y)) = x, then f and y are said to be inverse of each other.
Example: If f(x) = 2x + 5 = y, then, g(y) = (y-5)/2 = x is the inverse of f(x).
• The inverse of f is denoted by f -1

## Inverse Trigonometric Functions

• Trigonometric functions are many-one functions but we know that inverse of function exists if the function is bijective.
• If we restrict the domain of trigonometric functions, then these functions become bijective and the inverse of trigonometric functions are defined within the restricted domain.
• Example: y = f(x) = sin x, then its inverse is x = sin-1 y.

Inverse Trigonometric Formulas

Fig: Inverse Trigonometric Functions

The inverse trigonometric functions are the inverse functions of the trigonometric functions written as cos-1 x, sin-1 x, tan-1 x, cot-1 x, cosec-1 x, sec-1 x.

The inverse trigonometric functions are multivalued. For example, there are multiple values of ω such that z = sinω, so sin-1z is not uniquely defined unless a principal value is defined.

Such principal values are sometimes denoted with a capital letter so, for example, the principal value of the inverse sine may be variously denoted sin-1z or arcsinz.

Let’s say, if y = sin x , then x = sin-1 y, similarly for other trigonometric functions. This is one of the inverse trigonometric formulas. Now, y = sin-1 (x), y ∈ [π/2 , π/2] and x ∈ [-1,1].

• Thus, sin-1 x has infinitely many values for given x ∈ [-1, 1].
• There is only one value among these values which lies in the interval [π/2, π/2]. This value is called the principal value.

Domain and Range of Inverse Trigonometric Formulas

 Function Domain Range sin-1x [-1,1] [-π/2,π/2] cos-1x [-1,1] [0,π] tan-1x R (-π/2,π/2) cot-1x R (0,π) sec-1x R-(-1,1) [0,π]-{π/2} cosec-1x R-(-1,1) [-π/2,π/2]-{0}

Solved Examples

Ques 1: Find the exact value of each expression without a calculator, in [0,2π).

1. sin-1(−3√2)
2. cos-1(−2√2)
3. tan-1√3

Ans:

• Recall that −3√2 is from the 30−60−90 triangle. The reference angle for sin and 3√2 would be 60∘. Because this is sine and it is negative, it must be in the third or fourth quadrant. The answer is either 4π/3 or 5π/3.
• −2√2is from an isosceles right triangle. The reference angle is then 45∘. Because this is cosine and negative, the angle must be in either the second or third quadrant. The answer is either 3π/4 or 5π/4.
• √3is also from a 30−60−90 triangle. Tangent is √3 for the reference angle 60∘. Tangent is positive in the first and third quadrants, so the answer would be π/3 or 4π/3.

Notice how each one of these examples yields two answers. This poses a problem when finding a singular inverse for each of the trig functions. Therefore, we need to restrict the domain in which the inverses can be found.

Ques 2: Find the value of tan-1(1.1106).
Ans:
Let A=tan−1(1.1106)
Then, tanA = 1.1106
A = 48°
tan48 = 1.1106
[Use calculator in degree mode]
tan−1 1.1106=48°

Properties of Inverse

Here are the properties of the inverse trigonometric functions with proof.

Property 1

1. sin-1 (1/x) = cosec-1x , x ≥ 1 or x ≤ -1
2. cos-1 (1/x) = sec-1x , x ≥ 1 or x ≤ -1
3. tan-1 (1/x) = cot-1x , x > 0

Proof : sin-1 (1/x) = cosec-1x , x ≥ 1 or x ≤ -1,

Let  sin−1x=y

i.e. x = cosec y

1/π = sin y

Property 2

1. sin-1(-x) = – sin-1(x),    x ∈ [-1,1]
2. tan-1(-x) = -tan-1(x),   x ∈ R
3. cosec-1(-x) = -cosec-1(x), |x| ≥ 1

Proof: sin-1(-x) = -sin-1(x),    x ∈ [-1,1]
Let,  Sin-1(-x) = y

Then −x=sin y

x=−sin y

x=sin(−y)

sin−1=sin−1(sin(−y))

sin−1x=y

sin−1x=−sin−1(−x)

Hence,sin−1(−x)=−sin−1 x ∈ [-1,1]

Property 3

1. cos-1(-x) = π – cos-1 x, x ∈ [-1,1]
2. sec-1(-x) = π – sec-1x, |x| ≥ 1
3. cot-1(-x) = π – cot-1x, x ∈ R

Proof : cos-1(-x) = π – cos-1 x, x ∈ [-1,1]

Let cos−1(−x)=y

cos y=−x   x=−cos y

x=cos(π−y)

Since,  cos π−q=−cos q

cos−1 x=π−y

cos−1 x=π–cos−1–x

Hence, cos−1−x=π–cos−1x

Solved Example

Ques 1: Prove that “sin-1(-x) = – sin-1(x),    x ∈ [-1,1]”

Ans: Let, sin−1(−x)=y

Then −x=sin y

x=−sin y

x=sin(−y)

sin−1 x=arcsin(sin(−y))

sin−1 x=y

sin−1 x=−sin−1(−x)

Hence, sin−1(−x)=−sin−1 x, x ∈ [-1,1]

This concludes our discussion on the topic of trigonometric inverse functions.
Ques 2: sin-1(cos π/3)=?
Ans:
sin−1  [substitute cos(π/3)=1/2]
= π/6 [substitute sin-1 (1/2) = π/6]

Ques 3: Find the value of sin (π/4+Cos-1(√2/2)).
Ans:

Let y=sin
Then, cosA= √2/2
Multiplying the numerator as well as denominator by √2 we get:

cosA=1/√2
A = π/4
Therefore
y = sin
y = sin (π/2)
hence, y=1.

The document Basics Concepts: Inverse Trigonometric Functions | Mathematics (Maths) for JEE Main & Advanced is a part of the JEE Course Mathematics (Maths) for JEE Main & Advanced.
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## Mathematics (Maths) for JEE Main & Advanced

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## FAQs on Basics Concepts: Inverse Trigonometric Functions - Mathematics (Maths) for JEE Main & Advanced

 1. What is the definition of an inverse function?
Ans. An inverse function is a function that undoes the effect of another function. If a function f(x) maps an input x to an output y, then its inverse function, denoted as f^(-1)(x), maps the output y back to the input x.
 2. How do inverse trigonometric functions relate to trigonometric functions?
Ans. Inverse trigonometric functions are used to find the angle or arcsine, arccosine, or arctangent when the ratio of sides in a right triangle is given. These functions "undo" the trigonometric functions by finding the angle or arc corresponding to a given ratio.
 3. What are some examples of inverse trigonometric functions?
Ans. Some examples of inverse trigonometric functions include arcsin(x), arccos(x), and arctan(x). These functions find the angle whose sine, cosine, or tangent is equal to the given value of x, respectively.
 4. How can inverse trigonometric functions be useful in real-life applications?
Ans. Inverse trigonometric functions are widely used in fields such as physics, engineering, and navigation. They help in solving problems involving angles and sides of triangles, such as finding the height of a building using trigonometric ratios and measurements.
 5. Are there any restrictions on the domains of inverse trigonometric functions?
Ans. Yes, there are restrictions on the domains of inverse trigonometric functions. The domain of arcsin(x) is -1 ≤ x ≤ 1, the domain of arccos(x) is -1 ≤ x ≤ 1, and the domain of arctan(x) is -∞ < x < ∞. These restrictions ensure that the output of the inverse trigonometric functions is a valid angle or arc.

## Mathematics (Maths) for JEE Main & Advanced

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