Inverse Circular functions

In summary, inverse circular functions are mathematical functions that are used to find the angle or arc length of a right triangle given the ratio of its sides. They are also known as arc functions or trigonometric inverses. The six inverse circular functions are arcsine, arccosine, arctangent, arccosecant, arcsecant, and arccotangent, represented by sin^-1, cos^-1, tan^-1, csc^-1, sec^-1, and cot^-1 respectively. To find the inverse of a circular function, you can use inverse trigonometric identities or a calculator with inverse function keys. Inverse circular functions have a domain that is the range of their corresponding circular functions and a
  • #1
himanshu121
653
1
I don't know how will i get these Limits for inverse Trigonometric functions for eg

[tex]2 \sin^{-1}x = - \pi - sin^{-1} [2x \sqrt{1-x^2}] for x \leq -\frac{1}{\sqrt{2}} [/tex]
=[tex] sin^{-1} [2x \sqrt{1-x^2}] -\frac{1}{\sqrt{2}} \leq x \leq \frac{1}{\sqrt{2}} [/tex]
=[tex] \pi - sin^{-1} [2x \sqrt{1-x^2}] x \geq \frac{1}{\sqrt{2}} [/tex]


I want to know how we arrive at these values Or intervals
 
Mathematics news on Phys.org
  • #2
The easiset way, I think, to do these kinds of problems is to (reversibly) convert it from an inverse trig function identity to a trig function identity.
 
  • #3
for x

To understand how these limits for inverse trigonometric functions are derived, we first need to understand the concept of inverse circular functions. Inverse circular functions are the inverse operations of circular functions (such as sine, cosine, and tangent). They are used to find the angle measure given the ratio of sides in a right triangle.

To find the limits for inverse trigonometric functions, we need to consider the domain and range of the original function. For example, in the case of 2 \sin^{-1}x, the domain of the function is -1 \leq x \leq 1. This means that the possible values of x for which the function is defined are between -1 and 1.

Now, let's look at the different intervals for x in the given problem. For x \leq -\frac{1}{\sqrt{2}}, the value of 2x \sqrt{1-x^2} is negative. This means that when we take the inverse sine of this value, we will get a negative angle. However, the domain of inverse sine is between -\frac{\pi}{2} and \frac{\pi}{2}. So, in order to get a valid angle measure, we need to add a negative sign to the inverse sine value, giving us - \pi - sin^{-1} [2x \sqrt{1-x^2}].

Similarly, for x \geq \frac{1}{\sqrt{2}}, the value of 2x \sqrt{1-x^2} is positive, and we can directly take the inverse sine without adding a negative sign. This gives us \pi - sin^{-1} [2x \sqrt{1-x^2}].

For the middle interval, -\frac{1}{\sqrt{2}} \leq x \leq \frac{1}{\sqrt{2}}, the value of 2x \sqrt{1-x^2} is between -1 and 1, which is the domain of inverse sine. Therefore, we can directly take the inverse sine without any additional modifications.

In summary, the limits for inverse trigonometric functions are derived by considering the domain and range of the original function and making adjustments to the angle measure to ensure that it falls within the valid domain of inverse circular functions.
 

What are inverse circular functions?

Inverse circular functions are mathematical functions that are used to find the angle or arc length of a right triangle given the ratio of its sides. They are also known as arc functions or trigonometric inverses.

What are the six inverse circular functions?

The six inverse circular functions are arcsine, arccosine, arctangent, arccosecant, arcsecant, and arccotangent. They are represented by sin-1, cos-1, tan-1, csc-1, sec-1, and cot-1, respectively.

How do you find the inverse of a circular function?

To find the inverse of a circular function, you can use the inverse trigonometric identities or use a calculator with inverse function keys. For example, to find the inverse of sine, use the arcsine function on your calculator or use the identity sin-1(x) = arcsin(x).

What is the domain and range of inverse circular functions?

The domain of inverse circular functions is the range of their corresponding circular functions. In other words, the domain of arcsine, arccosine, and arctangent is [-1, 1], while the domain of arccosecant, arcsecant, and arccotangent is (-∞, -1]U[1, ∞). The range of inverse circular functions is the domain of their corresponding circular functions.

How are inverse circular functions used in real-life applications?

Inverse circular functions are used in various fields such as engineering, physics, and navigation to solve problems involving right triangles. They are also used in computer graphics and animation to create smooth curves and movements. In addition, they are used in astronomy to calculate the positions of celestial objects.

Similar threads

Replies
2
Views
1K
  • General Math
Replies
5
Views
905
Replies
17
Views
2K
Replies
2
Views
1K
  • General Math
Replies
11
Views
1K
Replies
4
Views
788
Replies
2
Views
628
  • General Math
Replies
1
Views
128
Replies
3
Views
662
Replies
7
Views
1K
Back
Top