Problem 73
Question
Verify the Identity. $$\arcsin (-x)=-\arcsin x$$
Step-by-Step Solution
Verified Answer
The identity \(\arcsin(-x) = -\arcsin(x)\) is verified by trigonometric properties.
1Step 1: Introduction to Identity Verification
We are given the identity \( \arcsin (-x) = -\arcsin x \). Our goal is to verify whether this identity holds true for valid values of \( x \). The arcsine function, \( \arcsin(x) \), is defined as the inverse of the sine function for \( x \) in the range \(-1 \leq x \leq 1 \), and gives angles in the range \(-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}\).
2Step 2: Understanding the Functions
When we apply the arcsine to \(-x\), it will provide the angle whose sine is \(-x\). Since sine is an odd function, \( \sin(-x) = -\sin(x) \), the arcsine of \(-x\) should correspond to the negative of the arcsine of \(x\).
3Step 3: Verification via Trigonometric Properties
Consider \(\theta = \arcsin(x)\) which implies \(\sin(\theta) = x\). Since \(\sin(-\theta) = -\sin(\theta) = -x\), it follows that \(-\theta = \arcsin(-x)\). This confirms that \( \arcsin(-x) = -\arcsin(x) \) due to the property that \( \arcsin(x) \) is the angle that returns \(x\) when passed through sine.
Key Concepts
Arcsine FunctionTrigonometric IdentitiesOdd Functions
Arcsine Function
The arcsine function, denoted as \( \arcsin(x) \), is a fundamental inverse trigonometric function. It allows us to find the angle whose sine is \( x \). For example, if \( \sin(\theta) = x \), then \( \theta = \arcsin(x) \). Hence, the arcsine essentially "undoes" the sine function.
It's important to know the range of values for which the arcsine function is defined. The input, \( x \), must fall within the range of \(-1 \leq x \leq 1\). This is because sine values lie within this interval. The output, \( \theta \), is produced in the range of \(-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2} \).
Understanding arcsine is important to verify identities like \( \arcsin(-x) = -\arcsin(x) \), since it reflects the symmetry and properties of sine and its inverse.
It's important to know the range of values for which the arcsine function is defined. The input, \( x \), must fall within the range of \(-1 \leq x \leq 1\). This is because sine values lie within this interval. The output, \( \theta \), is produced in the range of \(-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2} \).
Understanding arcsine is important to verify identities like \( \arcsin(-x) = -\arcsin(x) \), since it reflects the symmetry and properties of sine and its inverse.
- Acts as an inverse for sine.
- Input range: \(-1 \leq x \leq 1\).
- Output range: \(-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}\).
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that are true for every value of the variables involved.
In the context of our exercise, the identity \( \arcsin(-x) = -\arcsin(x) \) is backed by the properties of both sine and arcsine. The identity showcases how trigonometric functions that appear complex can often be simplified or related through these identities.
Understanding identities helps in simplifying expressions and solving complex equations.
In the context of our exercise, the identity \( \arcsin(-x) = -\arcsin(x) \) is backed by the properties of both sine and arcsine. The identity showcases how trigonometric functions that appear complex can often be simplified or related through these identities.
Understanding identities helps in simplifying expressions and solving complex equations.
- Aids in linking different trigonometric relationships.
- Helps in verifying and simplifying expressions.
Odd Functions
Odd functions are a category of functions characterized by their symmetry about the origin.
A function \( f(x) \) is odd if it holds that \( f(-x) = -f(x) \). The sine function is an example of an odd function, which implies \( \sin(-x) = -\sin(x) \).
In the verification of the identity \( \arcsin(-x) = -\arcsin(x) \), the property of the sine function being odd plays a crucial role. This property essentially tells us that finding the sine of a negative angle results in the negative sine of the angle's positive counterpart.
A function \( f(x) \) is odd if it holds that \( f(-x) = -f(x) \). The sine function is an example of an odd function, which implies \( \sin(-x) = -\sin(x) \).
In the verification of the identity \( \arcsin(-x) = -\arcsin(x) \), the property of the sine function being odd plays a crucial role. This property essentially tells us that finding the sine of a negative angle results in the negative sine of the angle's positive counterpart.
- Defined by \( f(-x) = -f(x) \).
- Includes functions like sine.
- Essential for understanding certain trigonometric identities.
Other exercises in this chapter
Problem 72
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