Problem 9 - Algebra 2 and Trigonometry

Question

A function \(\mathrm{f}\) is odd if and only if \(\mathrm{f}(x)=-\mathrm{f}(-x)\) for all \(x\) in the domain of the function. Note that a function is odd if it is symmetric with respect to the origin. In other words, the function is its own image under a reflection about the origin. a. Draw a unit circle and any first-quadrant angle \(R O P\) in standard position, with point \(P\) on the unit circle. Let \(\mathrm{m} \angle R O P=\theta .\) b. On the same set of axes, draw an angle in standard position with measure \(-\theta .\) What is the relationship between \(\theta\) and \(-\theta ?\) Between \(\sin \theta\) and \(\sin (-\theta) ?\) c. Repeat steps a and b for second- third-, and fourth-quadrant angles. Does \(\sin \theta=\sin\) \((-\theta)\) for second-, third-, and fourth-quadrant angles? Justify your answer. d. Does \(\sin \theta=-\sin (-\theta)\) for quadrantal angles? Explain. e. Do parts a-d show that \(y=\sin x\) is an odd function? Justify your answer.

Step-by-Step Solution

Answer

The function \( y = \sin x \) is an odd function as \( \sin(-x) = -\sin(x) \) for all \( x \).

Step 1: Understanding Odd Function Definition

A function \( f(x) \) is considered an odd function if \( f(x) = -f(-x) \) for all \( x \) in its domain. This means the graph is symmetric with respect to the origin.

Step 2: Drawing the Unit Circle and Angle \( \theta \)

Draw a unit circle with a first-quadrant angle \( ROP \) where \( P \) lies on the unit circle. Let \( m \angle ROP = \theta \). This angle is measured from the positive x-axis.

Step 3: Analyzing Angle \(-\theta\)

On the same set of axes, draw the angle \(-\theta\), which reflects the angle \( \theta \) across the x-axis. Therefore, \( \sin(-\theta) = -\sin(\theta) \) due to the symmetry of the sine function relative to the origin.

Step 4: Extending to Other Quadrants

Repeat steps a and b for second-, third-, and fourth-quadrant angles. In each case, observe that \( \sin(-\theta) = -\sin(\theta) \). This is because reflecting \( \theta \) to \(-\theta\) still results in the negation of the sine value.

Step 5: Examining Quadrantal Angles

Check quadrantal angles (where \( \theta \) is \(0, \pi/2, \pi, 3\pi/2\)). For these angles too, \( \sin(-\theta) = -\sin(\theta) \) as reflecting across the x-axis results in the opposite sine value.

Step 6: Conclusion on the Sine Function

From the observations made in the previous steps, the function \( y = \sin x \) conforms to the definition of an odd function, since \( \sin(-x) = -\sin(x) \) for all angles \( x \). Thus, \( y = \sin x \) is an odd function.

Key Concepts

Odd Functions

Odd functions have a very specific mathematical definition. They satisfy the condition that for any given value of \( x \), \( f(x) = -f(-x) \). This directly reflects the function's symmetry around the origin. Think about it like a seesaw; if one point goes up, the corresponding point on the opposite side moves down equally. Because of this symmetry, the graph of an odd function will look like a mirror image around the origin when you fold the coordinate plane along the x and y axes.

Sine Function

The sine function, denoted \( \sin(\theta) \), is a crucial part of trigonometry and circles. Rooted deeply in the unit circle, this function describes how the y-coordinate changes as you move around the circle. The key takeaway about the sine function is its periodic nature and its odd function property. Its periodicity means it repeats values in consistent intervals. Meanwhile, its categorization as an odd function stems from the fact that \( \sin(-\theta) = -\sin(\theta) \). As you reflect an angle across the x-axis, this mirroring in the circle results in negating the original sine value, reinforcing its odd function characteristic.

Unit Circle

The unit circle is an essential tool in trigonometry. It's a circle with a radius of one unit centered at the origin of the coordinate plane. Each point on this circle represents an angle \( \theta \) measured from the positive x-axis. As you move around the circle, the x-coordinates are given by \( \cos(\theta) \) and the y-coordinates by \( \sin(\theta) \). Because of the circle's symmetry, relationships involving angles such as \(-\theta\) translate directly into reflections across the axis. This fundamental property is what makes the unit circle such a powerful tool for visualizing and understanding the behavior of trigonometric functions.

Angle Symmetry

Angle symmetry in trigonometry deals with how angles relate to each other through reflections or rotations. When you have an angle \( \theta \) and its negative counterpart \(-\theta\), they lie symmetrically on opposite sides of the x-axis. This reflects a key property of the sine function, where \( \sin(-\theta) = -\sin(\theta) \), demonstrating how angle symmetry causes a reversal of the sine's sign. This principle extends across all four quadrants of the unit circle, ensuring consistency and predictability in the sine function's values as you explore different rotational positions.