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Chapter 6: Problem 81

Verify the identity. $$\arcsin (-x)=-\arcsin x$$

Short Answer

Expert verified
\(\arcsin(-x) = -\arcsin(x)\) is verified using the odd function property of sine.

Step by step solution

01

Understand the Functions Involved

The exercise involves understanding the properties of the inverse sine function, known as \(\arcsin x\), which is the inverse of the sine function. The \(\arcsin x\) function returns the angle whose sine is \(x\) and ranges from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\). This identity checks if \(\arcsin (-x)\) equals \(-\arcsin x\).
02

Explore \(\arcsin(-x)\) Using Properties of Sine

We know that sine is an odd function, which means \(\sin(-\theta) = -\sin(\theta)\). Therefore, the angle \(\theta\) which satisfies \(\sin(\theta) = -x\) is the same angle which satisfies \(\sin(-\theta) = x\).
03

Establish the Angle Relationship

Since \(\arcsin(x)\) is the angle \(\theta\) such that \(\sin(\theta) = x\), and by the odd nature of sine, if \(\theta = \arcsin(x)\), then \(-\theta = \arcsin(-x)\).
04

Verify the Identity

The identity assumes \(\arcsin(-x) = -\arcsin(x)\). From Steps 2 and 3, we have shown that \(-\theta = \arcsin(-x)\) is indeed equivalent to \(-\arcsin(x)\). Thus, \(\arcsin(-x) = -\arcsin(x)\), verifying the identity.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inverse Trigonometric Functions
Inverse trigonometric functions play a vital role in trigonometry as they allow us to find angles when given trigonometric values. These functions essentially "reverse" the regular trigonometric functions. For example, the inverse sine function, written as \( \arcsin x \), gives the angle whose sine is \( x \). This function specifically helps in finding angles ranging from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \).
\( \arcsin(-x) = -\arcsin(x) \) is an important identity. It highlights the symmetry of the sine wave about the origin, thanks to the inverse function's responsiveness to sine's odd nature.
Sine Function Properties
The sine function has some specific properties that influence its graph and behavior. It is periodic, with a period of \( 2\pi \), meaning that the function repeats its values every \( 2\pi \) units. It is also continuous and smooth, making it easy to predict and work with mathematically.
Another essential property is that sine function is an odd function. This means for any angle \( \theta \), \( \sin(-\theta) = -\sin(\theta) \). This symmetry about the origin is what makes the identity \( \arcsin(-x) = -\arcsin(x) \) valid. It is crucial for understanding how sine behaves with negative inputs.
Odd Functions
An odd function is a function \( f \) where for every value \( x \), \( f(-x) = -f(x) \). This symmetry around the origin makes odd functions distinctive. The sine function is a classic example of such a function.
This means that if you take a point on the sine function graph and reflect it across the origin, you will land on another point on the graph. This property is reflected in inverse trigonometric identities like \( \arcsin(-x) = -\arcsin(x) \). Such identities make use of the sine function's odd nature which doesn't change when taking inverses.

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