Inverse Trigonometric Functions
Inverse trigonometric functions as a topic of learning are closely related to the basic trigonometric ratios. The domain (θ value) and the range(answer) of the trigonometric ratio are changed to the range and domain of the inverse trigonometric functions. We have heard about function f, and the inverse of a function f-1 exists if f is a one-one function. But on observation, trigonometric functions are not one-one function. Further limiting these trigonometric functions to only its principal values, we have inverse trigonometric functions.
Inverse trigonometric functions have all the formulas of the basic trigonometric functions, which include the sum of functions, double and triple of a function. Here we shall try to understand the transformation of the trigonometric formulas to inverse trigonometric formulas.
Inverse Trigonometric Functions-Introduction
Inverse trigonometric functions are the inverse ratio of the basic trigonometric ratios. Here the basic trigonometric function of Sin θ = x, can be changed to Sin-1 x = θ. Here x can have values in whole numbers, decimals, fractions, or exponents. For θ = 30° we have θ = Sin-1 (1/2). All the trigonometric formulas can be transformed into inverse trigonometric function formulas.
The below graphs help us to compare and understand the functions y = Sin-1x, and y = Cos-1x. The domain(x value) of the function is presented along the x-axis and the range(y value) of the inverse trigonometric function is presented along the y-axis.
The below graphs help us to compare and understand the functions y = Tan-1x, and y = Cot-1x. The domain(x value) of the function is presented along the x-axis and the range(y value) of the inverse trigonometric function is presented along the y-axis.
The below graphs help us to compare and understand the functions y = Sec-1x, and y = Cosec-1x. The domain(x value) of the function is presented along the x-axis and the range(y value) of the inverse trigonometric function is presented along the y-axis.
Further all the basic trigonometric function formulas have been transformed to the inverse trigonometric function formulas and are classified here as the following four sets of formulas.
- Arbitrary Values
- Reciprocal and Complementary functions
- Sum and difference of functions
- Double and triple of a function
Inverse Trigonometric Functions for Arbitrary Values
The inverse trigonometric ratio formula for arbitrary values is applicable for all the six trigonometric functions. For the inverse trigonometric functions of sine, tangent, cosecant, the negative of the values are translated as the negatives of the function. And for functions of cosecant, secant, cotangent, the negatives of the domain are translated as the subtraction of the function from the π value.
- Sin-1(-x) = -Sin-1x
- Tan-1(-x) = -Tan-1x
- Cosec-1(-x) = -Cosec-1x
- Cos-1(-x) = π - Cos-1x
- Sec-1(-x) = π - Sec-1x
- Cot-1(-x) = π - Cot-1x
Inverse Trigonometric Functions for Reciprocal and Complementary Functions
The inverse trigonometric functions for reciprocal and complementary functions are similar to the basic trigonometric functions. The reciprocal relationship of the basic trigonometric functions, sine-cosecant, cos-secant, tangent-cotangent, can be interpreted for the inverse trigonometric functions. Also the complementary functions, since-cosine, tangent-cotangent, and secant-cosecant can be interpreted into:
The inverse trigonometric formula of inverse sine, inverse cosine, and inverse tangent can also be expressed in the following forms.
- Sin-1x = Cosec-11/x
- Cos-1x = Sec-11/x
- Tan-1x = Cot-11/x
The complementary functions of sine-cosine, tangent-cotangent, secant-cosecant, sum up to π/2.
- Sin-1x + Cos-1x = π/2
- Tan-1x + Cot-1x = π/2
- Sec-1x + Cosec-1x = π/2
Sum and Difference of Inverse Trigonometric Functions
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), Tan(A + B). These inverse trigonometric function formulas can be used to further derive the double and triple of the functions.
- 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)
- Tan-1x + Tan-1y = Tan-1(x - y)/(1 + xy)
Double and Triple of Inverse Trigonometric Functions
The double and triple of inverse trigonometric functions have been derived from the basic trigonometric formulas.
Double of the Function
- 2Sin-1x = Sin-1(2x√(1 - x2))
- 2Cos-1x = Cos-1(2x2 - 1)
- 2Tan-1x = Tan-12x/(1 - x2)
Triple of the Function
- 3Sin-1x = Sin-1(3x - 4x3)
- 3Cos-1x = Cos-1(4x3 - 3x)
- 3Tan-1x = Tan1(3x - x3)/(1 - 3x2)
- Trigonometric Ratios
- Pythagoras Theorem
- Inverse Operations
- Pythagorean Theorem Calculator
Tips and Tricks on Inverse Trigonometric Functions
Some of the below tips would be helpful in solving and apply the various formulas of inverse trigonometric functions.
- Sin-1(Sinx) = Sin(Sin-1x) = x
- Sin-1x is different from (Sinx)-1. Also (Sinx)-1 = 1/Sinx
- Sin-1x = θ and θ refer to the angle, which is the principal value of this inverse trigonometric function.
Solved Examples on Inverse Trigonometric Functions
Example 1: Find the value of Tan-1(1) + Cos-1(-1/2) + Sin-1(-1/2).
Tan-1(1) + Cos-1(-1/2) + Sin-1(-1/2)
= π/4 + π - Cos-1(1/2) - Sin-1(1/2)
= π/4 + π - π/3 - π/6
= π/4 + π - π/2
= π/4 + π/2
Answer: Therefore the answer is 3π/4.
Example 2: Find the value of Tan-1(√3) - Cot-1(-√3).
Tan-1(√3) - Cot-1(-√3)
=Tan-1(√3) - (π - Cot-1(√3))
=Tan-1(√3) - π + Cot-1(√3)
= π/3 - π + π/6
= π/2 - π
Answer: Therefore the answer is -π/2.
FAQs on Inverse Trigonometric Functions
How Do You Find Inverse Trigonometric Function?
The inverse trigonometric functions of inverse sine, inverse cosine, or inverse tangent can be found from the basic trigonometric ratios.
Sin θ = x and θ = Sin−1x
What Is Inverse Trigonometry?
The inverse Trigonometry function formula helps to find the angle for the given inverse trigonometric function of the sides of a right-angled triangle (Sin−1x = θ)
What Are the Six Inverse Trigonometric Functions?
The six inverse trigonometric functions are:
Sin−1x, Cos−1x, Tan−1x, Cot−1x, Sec−1x, and Cosec−1x
How Do you Convert the Inverse Trigonometric Function of Sine Inverse to Tan Inverse?
The sine inverse is converted to tan inverse, as per the formula below.
What Is the Inverse of Sine Called?
The inverse sine is equal to cosecant.
Sinθ = 1 / Cosecθ
What Are the 3 Basic Trigonometric Functions?
The three basic trigonometric functions are Sinθ, Cosθ, and Tanθ.
Which Trigonometric Ratios Are Even?
The Trigonometric ratios of cosine and secant are even.
Cos(-θ) = Cosθ
Sec(-θ) = Secθ
What Are Arcsine, Arccosine, and Arctangent?
The terms arcsine, arccosine, and arctangent are the inverse ratio of the trigonometric ratios Sinθ, Cosθ, and Tanθ.
θ = sin-1x
θ = cos-1y