Inverse Circular Functions: Problems based on inverse trigonometric functions of trigonometric functions
Inverse Circular Functions: Problems based on inverse trigonometric functions of trigonometric functions
Inverse circular functions are the inverse functions of the circular trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent). These functions allow us to find the angle that corresponds to a given value of a trigonometric function.
In this topic, we will explore problems that involve finding the inverse of trigonometric functions and using them to solve equations and find angles.
Table of Inverse Circular Functions
| Trigonometric Function | Inverse Function | Domain | Range |
|---|---|---|---|
| Sine ($\sin$) | Arcsine ($\arcsin$) | $[-1, 1]$ | $[-\frac{\pi}{2}, \frac{\pi}{2}]$ |
| Cosine ($\cos$) | Arccosine ($\arccos$) | $[-1, 1]$ | $[0, \pi]$ |
| Tangent ($\tan$) | Arctangent ($\arctan$) | $(-\infty, \infty)$ | $(-\frac{\pi}{2}, \frac{\pi}{2})$ |
| Cosecant ($\csc$) | Arccosecant ($\arccsc$) | $(-\infty, -1] \cup [1, \infty)$ | $[-\frac{\pi}{2}, -\frac{\pi}{2}] \cup [\frac{\pi}{2}, \frac{\pi}{2}]$ |
| Secant ($\sec$) | Arcsecant ($\arcsec$) | $(-\infty, -1] \cup [1, \infty)$ | $[0, \frac{\pi}{2}] \cup [\pi, \frac{3\pi}{2}]$ |
| Cotangent ($\cot$) | Arccotangent ($\arccot$) | $(-\infty, \infty)$ | $(0, \pi)$ |
Important Points
- The domain of the inverse circular functions is the range of the corresponding circular trigonometric function.
- The range of the inverse circular functions is the domain of the corresponding circular trigonometric function.
- The inverse circular functions are denoted by adding "arc" or "arc-" as a prefix to the trigonometric function. For example, the inverse of sine is arcsine, the inverse of cosine is arccosine, and so on.
- The inverse circular functions are also sometimes denoted by using the notation $\sin^{-1}$, $\cos^{-1}$, $\tan^{-1}$, $\csc^{-1}$, $\sec^{-1}$, and $\cot^{-1}$.
- The values of the inverse circular functions are angles measured in radians.
- The inverse circular functions have restricted ranges to ensure that they are one-to-one functions and have unique values for each input.
Examples
Example 1: Finding the angle using the arcsine function
Find the angle $\theta$ that satisfies $\sin(\theta) = \frac{1}{2}$.
To find the angle, we can use the arcsine function. The arcsine function, denoted as $\arcsin$, is the inverse of the sine function.
Using the given equation, we have:
$$\sin(\theta) = \frac{1}{2}$$
Taking the inverse sine of both sides, we get:
$$\arcsin(\sin(\theta)) = \arcsin\left(\frac{1}{2}\right)$$
Since the arcsine function is the inverse of the sine function, the two functions cancel each other out, and we are left with:
$$\theta = \arcsin\left(\frac{1}{2}\right)$$
To find the value of $\theta$, we can use a calculator or reference table for the arcsine function. In this case, we find that $\arcsin\left(\frac{1}{2}\right) = \frac{\pi}{6}$.
Therefore, the angle $\theta$ that satisfies $\sin(\theta) = \frac{1}{2}$ is $\theta = \frac{\pi}{6}$.
Example 2: Solving an equation using the arccosine function
Solve the equation $\cos(x) = -\frac{1}{2}$ for $x$.
To solve this equation, we can use the arccosine function, denoted as $\arccos$, which is the inverse of the cosine function.
Using the given equation, we have:
$$\cos(x) = -\frac{1}{2}$$
Taking the inverse cosine of both sides, we get:
$$\arccos(\cos(x)) = \arccos\left(-\frac{1}{2}\right)$$
Since the arccosine function is the inverse of the cosine function, the two functions cancel each other out, and we are left with:
$$x = \arccos\left(-\frac{1}{2}\right)$$
To find the value of $x$, we can use a calculator or reference table for the arccosine function. In this case, we find that $\arccos\left(-\frac{1}{2}\right) = \frac{2\pi}{3}$.
Therefore, the solution to the equation $\cos(x) = -\frac{1}{2}$ is $x = \frac{2\pi}{3}$.
Conclusion
Inverse circular functions are useful for finding angles that correspond to specific values of trigonometric functions. By understanding the properties and ranges of these functions, we can solve equations and find angles in various trigonometric problems. Remember to use the appropriate inverse circular function based on the given trigonometric function and always check the range of the inverse function to ensure the solution is valid.