Important Properties of Inverse Trigonometric Functions

Inverse Trigonometric Functions are identified as the inverse functions of some basic Trigonometric functions such as sine, cosine, tangent, secant, cosecant, and cotangent functions. Inverse Trigonometric Functions are also known as anti Trigonometric functions, arcus functions, and cyclometric functions. These Inverse Trigonometric Functions formulas enable us to find out any angles with any of the Trigonometry ratios. These formulas are derived from the properties of Trigonometric functions.Through this article, we learn about Inverse Trigonometry concepts, properties of Inverse Trigonometric Functions, Inverse Trigonometric Functions problems etc.

The Inverse Trigonometric Functions' range and domain are transformed from the domain and range of Trigonometric functions. We study the relationships between angles and sides in a right-angled triangle in Trigonometry. Inverse Trigonometry functions exist as well. Sin, cos, tan, cosec, sec, and cot are the basic Trigonometric functions. Inverse Trigonometric Functions are symbolized by sin-1x, cos-1x, cot-1x, tan-1x, cosec-1x, and sec-1x, respectively.

Inverse Trigonometry Formulas

The following formulas have been combined together to provide a list of Inverse Trigonometric formulas. These formulas can be used to convert one function to another, calculate the functions' principal angle values, and execute a variety of arithmetic operations across these Inverse Trigonometric Functions. In addition, all of the basic Trigonometric function formulas have been converted to Inverse Trigonometric Function formulas and are organized into the four groups below.

• Arbitrary Values

• Reciprocal and Complementary functions

• Sum and difference of functions

• Double and triple of a function

Inverse Trigonometric Functions also termed as “Arc functions” or anti Trigonometric functions are the inverse functions of some basic Trigonometric functions. They are used to find out the unknown values of angles of a right- angle triangle with any of the Trigonometry ratios. We know that Trigonometric functions are usually applied to the right angle triangle. They are widely used in the field of engineering, physics, geometry and navigation. The six Trigonometric functions are used to find out the angle measurement of a right angle triangle when the measurements of  two sides of the triangle are known.

There are six Trigonometric functions for each of the Trigonometry ratios. The inverse of those six Trigonometric functions are:

• Arcsine

• Arccosine

• Arctangent

• Arcsecant

• Arcosecant

• Arccontangent

Properties of Inverse Trigonometry

Inverse Trigonometric Functions have qualities that are determined by the domain and range of the functions. There are a few aspects of Inverse Trigonometric Functions that are essential not only for solving issues but also for gaining a better knowledge of this topic. The range of values that an inverse function can achieve with the defined domain of the function is defined as the range of values that the inverse function can achieve. A function's domain is defined as the collection of all conceivable independent variables in which the function can exist. Inverse Trigonometric Functions have a fixed range of values.

Property 1:

• Sin−1(x) = cosec−1(\[\frac{1}{x}\]), x ∈ [−1,1]−{0}

• Cos−1(x) = sec−1(\[\frac{1}{x}\]), x ∈ [−1,1]−{0}

• Tan−1(x) = cot−1(\[\frac{1}{x}\]), if x > 0  (or)  cot−1(\[\frac{1}{x}\]) −π, if x < 0

• Cot−1(x) = tan−1(\[\frac{1}{x}\]), if x > 0 (or) tan−1(\[\frac{1}{x}\]) + π, if x < 0

Property  2: 

• Sin−1(−x) = −Sin−1(x)

• Tan−1(−x) = −Tan−1(x)

• Cos−1(−x) = π − Cos−1(x)

• Cosec−1(−x) = − Cosec−1(x)

• Sec−1(−x) = π − Sec−1(x)

• Cot−1(−x) = π − Cot−1(x)

Property 3: 

• Sin−1(\[\frac{1}{x}\]) = cosec−1x, x≥1 or x≤−1

• Cos−1(\[\frac{1}{x}\]) = sec−1x, x≥1 or x≤−1

• Tan−1(\[\frac{1}{x}\]) = −π + cot−1(x)

Property 4: 

• Sin−1(cos θ) = π/2 − θ, if θ ∈ [0,π]

• Cos−1(sin θ) = π/2 − θ, if θ ∈ [−π/2, π/2]

• Tan−1(cot θ) = π/2 − θ, θ ∈ [0,π]

• Cot−1(tan θ) = π/2 − θ, θ ∈ [−π/2, π/2]

• Sec−1(cosec θ) = π/2 − θ, θ ∈ [−π/2, 0] ∪ [0, π/2]

• Cosec−1(sec θ) = π/2 − θ, θ ∈ [0,π]−{π/2}

• Sin−1(x) = cos−1 \[\sqrt{(1− x^2)}\], 0 ≤ x ≤ 1

Property 5:

• Sin−1x + Cos−1x = π/2

• Tan−1x + Cot−1(x) = π/2

• Sec−1x + Cosec−1x = π/2

Solved Examples

1. Prove the equation below: “Sin-1 (-x) = - Sin-1 (x), x ϵ (-1, 1)”

Solution: 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

Hence, Sin-1 (-x) = - Sin-1 (x), x ϵ (-1, 1)

2. Prove that  Cos-1 (4x3 - 3 x) = 3 Cos-1 x , ½ ≤ x ≤ 1.

Solution: Let x = Cos ϴ

Where ϴ = Cos-1 (-x)

LHS = Cos-1 (-x) (4x3 -3x)

By substituting the value of x, we get

= Cos-1 (-x) (4 Cos3ϴ - 3 Cosϴ)

Accordingly, we get,

Cos-1 (Cos 3ϴ) = 3ϴ

By substituting the value of ϴ, we get

= 3 Cos-1 x

= RHS

Hence proved

Quiz Time

1. What is the principal value of the expression Cos-1Cos(−680°) ?

a. 2 π/9

b. -2 π/9

c. π/9

d. 34 π/9

2. Tan-1 (sin- π/2) is equal to

a.  -1

b.  1

c.   π/2

d.  – π/4

Uses of Inverse Trigonometry

They are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and they are used to calculate an angle from any of its Trigonometric ratios. Engineering, navigation, physics, and geometry all require Inverse Trigonometric Functions.

If you're a carpenter, for example, and you want to make sure the end of a piece of wood molding is cut at a 45-degree angle, you can utilize Inverse Trigonometric Functions. You may determine the angle of the cut by measuring the side lengths at the end of the molding and using an Inverse Trigonometric Function. As a result, inverse trig functions can be employed for various carpentry-related operations such as construction.

Application of Inverse Trigonometry

In the fields of engineering, construction, and architecture, Inverse Trigonometric ratios are frequently used. Inverse Trigonometric ratios are the simplest approach to obtain an unknown angle, therefore we utilize them in locations where we need to know the angle for our assistance and rapidly acquire the desired result. The following are a few examples of Inverse Trigonometric ratios in use:

1. A right-angled triangle's unknown angles are measured using this formula.

2. Used to determine the depth of a hole or the angle of inclination.

3. It is used by architects to determine the angle of a bridge and its supports.

4. Carpenters use this tool to achieve a specific cut angle.

Facts

• Hipparchus is the father of Trigonometry who compiled the first Trigonometry table

• Inverse Trigonometric Functions were introduced early in 1700x by Daniel Bernouli used A, sin for the inverse sine of a number

• Euler wrote “A t” for the inverse tangent in 1736.