Problem 3
Question
Zeigen Sie mit Hilfe von \(-\sin x=\sin (-x)\), \(\operatorname{da} \beta\) aus \(0
\leq \sin x \leq x\) für \(0 \leq x<1\) folgt
\(0 \geq \sin x \geq x \quad\) fur \(-1
Step-by-Step Solution
Verified Answer
Using \(-\sin x = \sin(-x)\), the inequality for \(-1 < x < 0\) can be shown by applying transformations from \(0 < -x < 1\).
1Step 1: Understanding the Given Range
We are given two ranges, \(0 \leq x < 1\) and \(-1 < x < 0\). For \(0 \leq x < 1\), it is given that \(0 \leq \sin x \leq x\). Our goal is to show that for \(-1 < x < 0\), \(0 \geq \sin x \geq x\).
2Step 2: Using the Identity \\( -\sin x = \sin(-x)\\)
We know from trigonometric identities that \(-\sin x = \sin(-x)\). This helps us relate the values of sine for negative inputs to their positive counterparts. Particularly, if \(0 \leq \sin(-x) \leq -x\) for \(0 < -x < 1\), then \(-\sin x \leq -x\).
3Step 3: Analyze the Condition for Negative Range
If we take the inequality \(-\sin x = \sin(-x)\) and apply it to the second range \(0 < -x < 1\), it transforms to \(-x < 0\). Recognizing that \(-x\) fits the given criteria \(0 \leq \sin(-x) \leq -x\), by comparing both sides we arrange it to \(0 \geq \sin x \geq x\).
4Step 4: Conclude the Proof
With the synthesis of both trigonometric identity and rearrangement, we have derived \(0 \geq \sin x \geq x\) from \(-1 < x < 0\) as predicted. This satisfies the inequality condition using the provided range and identity.
Key Concepts
Sine FunctionInequalitiesMathematical Proofs
Sine Function
The sine function is a fundamental part of trigonometry and is essential for understanding how angles relate to dimensions in circles and oscillatory movements. The function itself, denoted as \( \sin(x) \), calculates the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle. It's noteworthy for its periodic nature, repeating its values in cycles of \( 2\pi \).
What makes sine particularly interesting is its behavior over different ranges. For angles between 0 and \(\pi/2\) (i.e., the first quadrant), the sine value increases from 0 to 1. This feature is useful for understanding any mapping between sine values and their angles, especially within constrained segments, like in the problem at hand.
Keep in mind:
What makes sine particularly interesting is its behavior over different ranges. For angles between 0 and \(\pi/2\) (i.e., the first quadrant), the sine value increases from 0 to 1. This feature is useful for understanding any mapping between sine values and their angles, especially within constrained segments, like in the problem at hand.
Keep in mind:
- \( \sin(x) \) remains positive for all x in \( (0, \pi) \).
- The sine function is odd, implying \( \sin(-x) = -\sin(x) \).
- Its value is limited between -1 and 1 for all \( x \).
Inequalities
Inequalities are statements about the relative size or order of two objects, using symbols like \( > , < , \geq , \text{and} \leq \). In mathematical analysis, inequalities help illustrate constraints or bounds in a problem rather than specific solutions. They're especially important in proofs and assertions about functions.
For instance, in the examined exercise, the inequality \( 0 \leq \sin x \leq x \) for \( 0 \leq x < 1 \) is critical. It indicates that the sine of \( x \) is always less than or equal to \( x \) within this range. This property arises due to the sine function's arc-like growth, capturing it under a straight line passing through the origin.
Remember:
For instance, in the examined exercise, the inequality \( 0 \leq \sin x \leq x \) for \( 0 \leq x < 1 \) is critical. It indicates that the sine of \( x \) is always less than or equal to \( x \) within this range. This property arises due to the sine function's arc-like growth, capturing it under a straight line passing through the origin.
Remember:
- Inequalities can help provide bounds on functions and establish general behavior, even if exact values are unknown.
- Transforming inequalities through algebraic manipulation often reveals deeper truths about a problem's structure.
Mathematical Proofs
Mathematical proofs cement the validity of statements or theorems through logical deduction. They're like a detective story where each step builds towards a final confirmed conclusion. Proofs rest on basic mathematical principles, axioms, or previously known facts, meticulously constructing the truth.
Recall our problem: We began with known conditions and a trigonometric identity to demonstrate the inequality for a different range. In doing so, this proof showcased several steps typical to mathematical proofs:
Recall our problem: We began with known conditions and a trigonometric identity to demonstrate the inequality for a different range. In doing so, this proof showcased several steps typical to mathematical proofs:
- Defining the range and what needs to be proven.
- Utilizing identities, such as \( -\sin x = \sin(-x) \), to connect various conditions.
- Systematically transforming conditions, leading each logical step towards the desired conclusion.