Sin(arcsin x) = x

Definition

The expression 'sin(arcsin x) = x' is a fundamental relationship in trigonometry that connects the inverse trigonometric function, arcsin (also known as the inverse sine function), with the sine function. This relationship is a key property that helps in understanding and working with inverse trigonometric functions, which are essential in various mathematical applications.

5 Must Know Facts For Your Next Test

1. The expression 'sin(arcsin x) = x' means that the sine of the angle whose sine is x is equal to x itself.
2. This relationship is a consequence of the fact that the arcsin function is the inverse of the sine function.
3. The domain of the sine function is all real numbers, while the domain of the arcsin function is the interval [-1, 1].
4. The range of the sine function is [-1, 1], and the range of the arcsin function is [-$\pi/2$, $\pi/2$].
5. This property is useful in simplifying expressions and solving problems involving inverse trigonometric functions.

Review Questions

• Explain the significance of the relationship 'sin(arcsin x) = x' in the context of inverse trigonometric functions.
  • The relationship 'sin(arcsin x) = x' is a fundamental property that highlights the inverse nature of the sine function and the arcsin (inverse sine) function. It means that if you take an angle whose sine is x, and then find the sine of that angle, the result will be x itself. This property is essential in understanding and working with inverse trigonometric functions, as it allows for simplifying expressions and solving problems involving these functions. It demonstrates the 'undoing' nature of the inverse operation, where applying the inverse function to the original function's output results in the original input.
• Describe how the domains and ranges of the sine function and the arcsin function are related to the property 'sin(arcsin x) = x'.
  • The property 'sin(arcsin x) = x' is closely tied to the domains and ranges of the sine function and the arcsin (inverse sine) function. The domain of the sine function is all real numbers, while the domain of the arcsin function is the interval [-1, 1]. This is because the sine function can take on any real value between -1 and 1, and the arcsin function is defined to find the angle whose sine is within this range. The range of the sine function is [-1, 1], and the range of the arcsin function is [-$\pi/2$, $\pi/2$]. The relationship 'sin(arcsin x) = x' holds true because the input to the sine function (the angle whose sine is x) is exactly the same as the output of the arcsin function, which is the angle whose sine is x.
• Analyze how the property 'sin(arcsin x) = x' can be used to simplify expressions and solve problems involving inverse trigonometric functions.
  • The property 'sin(arcsin x) = x' can be very useful in simplifying expressions and solving problems that involve inverse trigonometric functions, such as the arcsin function. For example, if you have an expression like 'sin(arcsin(2/3)),' you can immediately simplify it to '2/3' using the 'sin(arcsin x) = x' relationship. This property allows you to 'undo' the inverse function and directly obtain the original value. Similarly, when solving equations or inequalities that involve inverse trigonometric functions, you can use this property to isolate the variable of interest and find the solution. The ability to apply this fundamental relationship is crucial in working with inverse trigonometric functions and simplifying complex expressions that arise in various mathematical contexts.

Related terms

Inverse Trigonometric Functions:

Inverse trigonometric functions are the inverse operations of the standard trigonometric functions (sine, cosine, tangent, etc.). They allow us to find the angle given the value of the trigonometric function.

Arcsin (Inverse Sine): The arcsin (or inverse sine) function is the inverse of the sine function. It takes a value between -1 and 1 and returns the angle whose sine is that value.

Domain and Range:

The domain of a function is the set of input values for which the function is defined, while the range is the set of output values the function can produce.