Problem 49 - Precalculus: A Concise Course

Question

Use the properties of inverse trigonometric functions to evaluate the expression. $$ \sin (\arcsin 0.3) $$

Step-by-Step Solution

Answer

The evaluated value of the expression \( \sin(\arcsin 0.3) \) is 0.3.

Step 1: Understanding the problem

The expression deals with the function \( \sin(\arcsin x) \). This is the composition of a function and its inverse, which yields the original input. That means, for any x in the domain of sine, it holds that \( \sin(\arcsin x) = x \).

Step 2: Application of the Property

Applying this property to the given expression \( \sin(\arcsin 0.3) \) will simplify it to 0.3.

Key Concepts

Trigonometric Identities

When understanding inverse trigonometric functions, it's crucial to grasp the related trigonometric identities. Trigonometric identities are equations involving trigonometric functions that are universally true. For instance, one such identity is the composition of a trigonometric function and its inverse. This identity states that if you take a function, apply its inverse to some value, and then apply the original function again, you end up with the original value (as long as it is within the proper range). This can be mathematically expressed as \( \sin(\arcsin(x)) = x \) for values of \( x \) within the domain of \( \arcsin \).

• These identities are extremely useful because they simplify complex expressions.
• They help to establish relationships between different trigonometric functions.
• Understanding these identities can make solving trigonometric equations much easier.

In the context of the example provided, the identity \( \sin(\arcsin(x)) = x \) directly simplifies \( \sin(\arcsin 0.3) \) to just 0.3.

Domain and Range

The concept of the domain and range is central to understanding inverse trigonometric functions. The domain of a function is the complete set of possible values of the independent variable. The range is the complete set of possible values of the dependent variable after applying the function. For inverse trigonometric functions like \( \arcsin(x) \), the domain is typically \(-1 \leq x \leq 1\). This is because these functions are designed to retrieve angles whose sine values are between -1 and 1.

• The range of \( \arcsin(x) \) is \(-\frac{\pi}{2} \leq y \leq \frac{\pi}{2} \), providing the principal value of the angle whose sine is \( x \).
• Understanding the domain and range helps ensure that we apply these functions correctly in trigonometric equations and identities.
• When dealing with \( \sin(\arcsin(x)) = x \), knowing the restrictions on \( x \) helps avoid any calculations that are mathematically invalid.

This is why in the expression \( \sin(\arcsin 0.3) \), \( 0.3 \) fits neatly within the domain of \( \arcsin \), making it a valid input.

Function Composition

Function composition involves the combination of two functions where the output of one function becomes the input of another. In the scenario where inverse trigonometric functions are involved, one often applies this idea to simplify expressions. For example, in \( \sin(\arcsin(x)) \), you are composing the functions \( \sin \) and \( \arcsin \).

• The essence of this composition is to "undo" the effect of the function by its inverse, simplifying the overall operation.
• This makes function composition an efficient tool when tackling problems involving inverse functions.
• It also reinforces understanding of how trigonometric and their inverse functions are interrelated.

In the given exercise, by applying this principle with \( \sin(\arcsin 0.3) \), we effectively bypass unnecessary calculations, directly reaching the result, 0.3. This seamless transition demonstrates one of the most straightforward applications of function composition.