Problem 4 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Suppose \(\tan \alpha=b\). Find the following. Explain your reasoning. (a) \(\tan (\alpha+\pi)\) (b) \(\tan (-\alpha)\) (c) \(\tan (\pi-\alpha)\)

Step-by-Step Solution

Answer

The solutions are \(\tan (\alpha+\pi) = b\), \(\tan (-\alpha) = -b\) and \(\tan (\pi-\alpha) = -b\).

Step 1: Calculate \(\tan (\alpha+\pi)\)

We know that the tangent function has a period of \(\pi\) or is periodic with a period of \(\pi\). This means that \(\tan(\theta + \pi) = \tan (\theta)\). So, \(\tan (\alpha + \pi) = \tan \alpha\). Given that \(\tan \alpha = b\), we can say that \(\tan (\alpha + \pi) = b\).

Step 2: Calculate \(\tan (-\alpha)\)

The tangent function is an odd function so \(\tan(-\theta) = -\tan \theta\). So, \(\tan (-\alpha) = -\tan \alpha\). Given \(\tan \alpha = b\), we find that \(\tan (-\alpha) = -b\).

Step 3: Calculate \(\tan (\pi - \alpha)\)

The tangent function has a property where \(\tan(\pi - \theta) = -\tan \theta\). Hence, \(\tan (\pi - \alpha) = -\tan \alpha\). Given \(\tan \alpha = b\), we get \(\tan (\pi - \alpha) = -b\).

Key Concepts

Periodicity

The tangent function exhibits a fascinating property called periodicity. Periodicity refers to how a function repeats its values at regular intervals. For the tangent function, this period is \(\pi\). This means that every time we add \(\pi\) to the angle, the value of the tangent function remains the same. In mathematical terms, it can be expressed as:

• \(\tan(\theta + \pi) = \tan(\theta)\)

Using this property, we can determine:

• \(\tan(\alpha + \pi) = \tan\alpha\)

So, if \(\tan\alpha = b\), then \(\tan(\alpha + \pi) = b\). This highlights how the function repeats itself after an interval of \(\pi\), making it predictable and useful in complex calculations.

Odd Function

An odd function is a type of mathematical function that exhibits a specific kind of symmetry. For odd functions, the function evaluated at the negative of an angle is equal to the negative of the function evaluated at the angle itself. In symbol form:

• \(f(-x) = -f(x)\)

The tangent function is an example of an odd function. This property is useful when understanding transformations and symmetries in trigonometry:

• For tangent: \(\tan(-\theta) = -\tan(\theta)\)

This means if \(\tan\alpha = b\), then \(\tan(-\alpha) = -b\). Recognizing this symmetry can simplify solving problems since you can instantly determine the function's behavior by flipping the sign of the angle.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all angle values. These identities are essential tools in simplifying and solving trigonometric problems. One vital identity related to tangent is:

• \(\tan(\pi - \theta) = -\tan(\theta)\)

Using this identity helps determine transformations involving specific angle shifts, such as:

• \(\tan(\pi - \alpha) = -\tan\alpha\)

Given that \(\tan\alpha = b\), we find \(\tan(\pi - \alpha) = -b\). These identities allow for straightforward calculations and provide deeper insights into relationships between angles and their tangent values.