Inverse Trigonometric Functions

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Introduction:

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It has numerous real-world applications in physics, engineering, and architecture. Trigonometric functions such as sine, cosine, and tangent describe these relationships.

Inverse trigonometric functions, also known as arc functions, are the inverse operations of trigonometric functions. In other words, they can be used to find the angle of a right triangle when given the ratio of its sides.

In this blog,we will learn about the following topics

  • What are Inverse functions (Brief Introduction)
  • What are Inverse Trigonometric Functions?
  • Understanding Trigonometric Functions and their Inverses
  • The Range and Domain of Inverse Trigonometric Functions
  • Graphs of Inverse Trigonometric Functions
  • Properties of Inverse Trigonometric Functions
  • Applications of Inverse Trigonometric Functions
  • Solving Trigonometric Equations Using Inverse Trigonometric Functions
  • Common Formulas and Identities for Inverse Trigonometric Functions
  • Tips for Working with Inverse Trigonometric Functions
  • Conclusion: Why Inverse Trigonometry Functions are Important.

So buckle up, and get ready to journey through the world of inverse trigonometric functions!

What are Inverse functions (Brief Introduction)

Suppose in a laboratory, the number of bacteria N varies with time according to the function-

$$N=f(t)$$

Then we will say N is the function of time t, but if a biologist is interested in the time required for the population of bacteria to reach various levels, then he wants a function something like

$$t=f^{-1}(N)$$

Here t is a function of N, It is called the Inverse function of N.

So, $$y=f(x)$$ which implies $$f^{-1}(y)=x$$

 

Note: Only those functions, which are one-to-one,  can have their inverse functions. To check whether any function is one-to-one or not, you can use the Horizontal line test.

What are Inverse Trigonometric Functions?

Inverse trigonometry functions are the inverse operations of the six primary trigonometric functions: sine, cosine, tangent, cosecant, secant, and cotangent. They are denoted by the prefix “arc” or “a” and the name of the trigonometric function. For example, the inverse sine function is denoted by $$\arcsin$$ or $$\sin^{-1}$$.

The inverse trigonometric functions are also referred to as arc functions.

Note:- $$sin^{-1}x$$ and $$(sin x)^{-1}$$ are two different things. $$Sin^{-1} x$$ is an inverse Trigonometric function while $$(sin x)^{-1}=\frac{1}{sin x}=cosec x$$

Understanding Trigonometric Functions and their Inverses

Trigonometric functions are defined in terms of the ratios of the sides of a right triangle. The sine function is defined as the ratio of the opposite side to the hypotenuse, the cosine function is defined as the ratio of the adjacent side to the hypotenuse, and the tangent function is defined as the ratio of the opposite side to the adjacent side.

On the other hand, inverse trigonometric functions are used to find the angle of a right triangle when given the ratios of its sides. For example, if we know the ratio of the opposite side to the hypotenuse, we can use the inverse sine function to find the angle opposite that side.

The Range and Domain of Inverse Trigonometric Functions

Since we know that only those functions have inverse functions which are one-to-one, we would have to restrict the domains of trigonometric functions to make them one-to-one.

Consider a sine function $$y=sin x$$

If we restrict its domain to interval $$\bigg[-\frac{π}{2},\frac{π}{2}\bigg]$$ then it’s all values(outputs) are attained.

In this way, all the trigonometric functions can become one-to-one. The domain and ranges of all inverse trigonometric functions are given below in the diagram-

The values of cosec x, sec x, and cot x are not defined at x=0, π/2, and 0, respectively, and their inverses have domains and range accordingly.

Graphs of Inverse Trigonometric Functions

The graphs of inverse trigonometry functions are important tools for visualizing their behavior. The graphs of the inverse sine, inverse cosine, inverse tangent, inverse secant, inverse cosecant, and cotangent functions are shown below:

Applications of Inverse Trigonometric Functions

Inverse trigonometry functions have a wide range of applications in various fields, including:

Engineering: Inverse trigonometry functions calculate the angles and distances in structures such as bridges and buildings.

Physics: Inverse trigonometry functions are used to calculate the direction and speed of moving objects.

Navigation: Inverse trigonometry functions determine the position and orientation of objects such as ships and planes.

Computer graphics: Inverse trigonometry functions create 3D animations and video games.

Solving Trigonometric Equations Using Inverse Trigonometric Functions

Inverse trigonometric functions can be used to solve trigonometric equations.

For example, consider the equation $$\sin x = 0.5$$. We can use the inverse sine function to find the solutions:

$$x=sin^{-1}(0.5)=sin^{-1}\bigg(\frac{1}{2}\bigg)=\frac{π}{6}$$

because $$sin\bigg(\frac{π}{6}\bigg)=\frac{1}{2} and \bigg(\frac{π}{6}\bigg)\in \bigg[-\frac{π}{2},\frac{π}{2}\bigg]$$

Common Formulas and Identities for Inverse Trigonometric Functions

  • $$sin^{-1}x=cosec^{-1}\bigg(\frac{1}{x}\bigg)$$
  • $$cos^{-1}x=sec^{-1}\bigg(\frac{1}{x}\bigg)$$
  • $$tan^{-1}x=cot^{-1}\bigg(\frac{1}{x}\bigg)$$
  • $$cosec^{-1}x=sin^{-1}\bigg(\frac{1}{x}\bigg)$$
  • $$sec^{-1}x=cos^{-1}\bigg(\frac{1}{x}\bigg)$$
  • $$cot^{-1}x=tan^{-1}\bigg(\frac{1}{x}\bigg)$$
  • $$\theta=sin^{-1}(sin\theta)=cos^{-1}(cos\theta)=tan^{-1}(tan\theta)$$
  • $$sin^{-1}(-x)=-sin^{-1}x$$
  • $$cos^{-1}(-x)=π-cos^{-1}x$$
  • $$tan^{-1}(-x)=-tan^{-1}x$$
  • $$sin^{-1}x=cos^{-1}\sqrt{1-x^2}=tan^{-1}\bigg(\frac{x}{\sqrt{1-x^2}}\bigg)$$
  • $$cos^{-1}x=sin^{-1}\sqrt{1-x^2}=tan^{-1}\bigg(\frac{\sqrt{1-x^2}}{x}\bigg)$$
  • $$tan^{-1}x=sin^{-1}\bigg(\frac{x}{\sqrt{1+x^2}}\bigg)=cos^{-1}\bigg(\frac{1}{\sqrt{1+x^2}}\bigg)$$
  • $$sin^{-1}x+cos^{-1}x=\frac{π}{2}$$
  • $$tan^{-1}x+cot^{-1}x=\frac{π}{2}$$
  • $$sec^{-1}x+cosec^{-1}x=\frac{π}{2}$$
  • $$tan^{-1}x \pm tan^{-1}y=tan^{-1}\bigg(\frac{x \pm y}{1 \mp xy}\bigg)$$
  • $$tan^{-1}x+tan^{-1}y+tan^{-1}z=tan^{-1}\bigg(\frac{x+y+z-xyz}{1-xy–yz-zx}\bigg)$$
  • $$sin^{-1}x \pm sin^{-1}y=sin^{-1}(x\sqrt{1-y^2} \pm y\sqrt{1-x^2})$$
  • $$cos^{-1}x \pm cos^{-1}y=cos^{-1}[xy \mp \sqrt{(1-x^2)}\sqrt{(1-y^2)}]$$
  • $$2tan^{-1}x=tan^{-1}\bigg(\frac{2x}{1-x^2}\bigg)=sin^{-1}\bigg(\frac{2x}{1+x^2}\bigg)=cos^{-1}\bigg(\frac{1-x^2}{1+x^2}\bigg)$$
  • $$2sin^{-1}x=sin^{-1}(2x\sqrt{1-x^2})$$
  • $$2cos^{-1}x=cos^{-1}(2x^2-1)$$
  • $$3sin^{-1}x=sin^{-1}(3x-4x^3)$$
  • $$3cos^{-1}x=cos^{-1}(4x^3-3x)$$
  • $$3tan^{-1}x=tan^{-1}\bigg(\frac{3x-x^3}{1-3x^2}\bigg)$$

Tips for Working with Inverse Trigonometric Functions

When working with inverse trigonometric functions, it’s important to keep the following tips in mind:

  • Make sure you’re working with the correct range and domain for the function you’re using.
  • Remember that inverse trigonometry functions are one-to-one, so there will be only one output value for each input value.
  • Be careful when using inverse trigonometric functions to solve equations, as there may be multiple solutions.
  • Practice using inverse trigonometric functions with various problems to familiarize yourself with their properties and applications.

Conclusion: Why Inverse Trigonometric Functions are Important.

Inverse trigonometry functions are important tools for solving problems that involve triangles and trigonometric functions. They have a wide range of applications in various fields, and understanding their properties and applications is essential for success in mathematics and related fields. Following the tips and guidelines outlined in this blog can make you more confident and proficient in using inverse trigonometric functions.

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About Colin Phillips
About Colin Phillips

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