Problem 9 - A Graphical Approach to Precalculus with Limits

Question

Use the even-odd identities to write of the following expressions as a trigonometric function of a positive number $$\sin (-0.5)$$

Step-by-Step Solution

Answer

\( \sin(-0.5) = -\sin(0.5) \)

Step 1: Identify the Even-Odd Identity

Recall that the sine function is an odd function. This means that for the sine function, the identity is: \( \sin(-x) = -\sin(x) \).

Step 2: Apply the Even-Odd Identity

Substitute \(-0.5\) for \(x\) in the odd identity for sine. This gives us: \( \sin(-0.5) = -\sin(0.5) \).

Step 3: Express as a Positive Number

Since \( -\sin(0.5) \) is equivalent to applying the odd function property to simplify \( \sin(-0.5) \) to a trigonometric function of a positive angle, we write the expression as \(-\sin(0.5)\).

Key Concepts

Even-Odd Identities

Understanding even-odd identities is crucial when working with trigonometric functions. These identities help us simplify expressions and solve equations by indicating how functions behave with negative inputs.

Here are a few key points to consider:

• Even functions have the property that their output does not change if the input is negated. Mathematically, this means for an even function, \( f(-x) = f(x) \).
• Odd functions, on the other hand, change their sign when their input is negated. That is, \( f(-x) = -f(x) \).
• Trigonometric functions like cosine and secant are even, while sine, tangent, cotangent, and cosecant are odd.

This means when you’re given a trigonometric expression with a negative angle, you can rely on these identities to transform it to an expression involving positive angles, often making it easier to work with.

Sine Function

The sine function is one of the core trigonometric functions. It is defined as the ratio of the opposite side to the hypotenuse in a right-angled triangle.

Here are critical aspects of the sine function:

• The sine function is periodic, with a period of \(2\pi\).
• It has a range between -1 and 1 inclusive, meaning the sine of any angle will always produce a result within this interval.
• Sine is an odd function, which aligns with the property \( \sin(-x) = -\sin(x) \).

Understanding these properties is essential because they allow us to work with different angle measures, simplifying the calculation of angles and helping to predict the behavior of the sine wave in various applications, from physics to engineering.

Odd Function

A function is termed "odd" due to its behavior concerning symmetry. This terminology is a nod to algebra where functions of odd degree exhibit similar symmetry.

Key aspects of odd functions include:

• An odd function is symmetric about the origin.
• The fundamental property of odd functions is \( f(-x) = -f(x) \), aiding in simplifying expressions and solving problems involving negative inputs.
• This property is prominent in trigonometry with functions like sine and tangent.

By leveraging the odd nature of these functions, we easily transition from negative to positive angles. This not only simplifies many calculations but also enhances understanding of trigonometric functions' behavior across different quadrants of the unit circle.