Let y=y-1 (x) be the inverse function of y = sinx

General
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let y=y-1 (x) be the inverse function of y = sinx

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What is the inverse function of y = \sin x?

Answer: To find the inverse function of y = \sin x, we need to understand how the inverse function works for trigonometric functions. The sine function is periodic, meaning it repeats its values in regular intervals, specifically every 2\pi. Therefore, the sine function isn’t one-to-one on its entire domain, which would make finding an inverse across all possible inputs impossible. Instead, we focus on a restricted domain where the sine function is one-to-one.

1. Understanding the Basic Concept

  • Function and Inverse: For a function f(x), an inverse function f^{-1}(x) reverses the operation of f(x). If y = f(x), then x = f^{-1}(y).

  • Restricting the Domain: For the sine function, we restrict the domain to [-\frac{\pi}{2}, \frac{\pi}{2}], where the function is increasing and hence one-to-one. On this interval, each y corresponds to exactly one x.

2. Finding the Inverse

  • From the Equation: Given y = \sin x, the inverse function could be expressed as x = \sin^{-1}(y) or x = \arcsin(y). This is valid only for y values in the range [-1, 1], since sine values lie between -1 and 1.

  • Domain and Range of Inverse: With y = \sin x restricted to [-\frac{\pi}{2}, \frac{\pi}{2}], the inverse function y = \sin^{-1}(x) has a domain [-1, 1] and range [-\frac{\pi}{2}, \frac{\pi}{2}].

3. Graphical Representation

  • Graph Features: The graph of y = \sin x from [-\frac{\pi}{2}, \frac{\pi}{2}] is a single arc. When finding the inverse, y = \sin^{-1}(x), the graph is reflected across the line y = x.

  • Symmetry and Reflection: The inverse graph will exhibit symmetry about the line y = x, effectively swapping the roles of the x and y coordinates of the original function.

4. Properties and Applications

  • Inverse Function Characteristics: Like all inverse functions, \sin^{-1}(x) will return to us the angle whose sine is x, providing outputs in the interval [-\frac{\pi}{2}, \frac{\pi}{2}].

  • Applications in Practice: The inverse sine function, \sin^{-1}(x), is often used in trigonometry when determining angles in a right triangle, particularly in solving equations involving \sin x = a.

5. Multiple Examples

  • Example 1: Solve for x: \sin x = \frac{1}{2}. Using the inverse, x = \sin^{-1}\left(\frac{1}{2}\right). This yields x = \frac{\pi}{6}, within the restricted domain.

  • Example 2: Solve for x: \sin x = 0. Using the inverse, x = \sin^{-1}(0). The output is x = 0, which is also within [-\frac{\pi}{2}, \frac{\pi}{2}].

6. Step-by-Step Calculation for Inverse

  1. Restrict the Domain: Limit the use of the function to -\frac{\pi}{2} to \frac{\pi}{2} for sine.
  2. Write the Original Equation: Begin with y = \sin x.
  3. Rearrange for the Inverse: Solve for x, giving x = \sin^{-1}(y) or x = \arcsin(y).
  4. Identify Domain/Range: Ensure values used for x fall within [-1, 1] and for y within [-\frac{\pi}{2}, \frac{\pi}{2}].
  5. Use in Equations/Applications: Substitute into solving equations for angles or interpreting problems.

7. Key Takeaways

  • Inverse Sin Function (\sin^{-1}(x)): This notation refers to the arcsine function, not the reciprocal of sine.
  • Valid Inputs: The arcsine can only take inputs from [-1, 1] due to the nature of the sine.
  • Output Values: The outputs or angle values of the arcsine function range from [-\frac{\pi}{2}, \frac{\pi}{2}], giving each sine value a unique corresponding angle within this range.

With this understanding, one can confidently determine angles using the inverse sine function, handle trigonometric equations involving sine, and readily apply arcsine principles to real-world contexts, such as in engineering or physics scenarios where angle measurement is essential.

@anonymous4