Problem 36 - A Graphical Approach to Precalculus with Limits

Question

Use identities to write each expression as a function with \(x\) as the only argument. $$\tan (2 \pi-x)$$

Step-by-Step Solution

Answer

\( \tan(2\pi - x) = -\tan x \).

Step 1: Apply the Identity for Tangent of a Difference

The expression \( \tan(2\pi - x) \) involves the tangent of a difference of angles. We can use the identity \( \tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b} \). For \( a = 2\pi \) and \( b = x \), it simplifies to \( \tan(2\pi - x) = \frac{\tan 2\pi - \tan x}{1 + \tan 2\pi \tan x} \).

Step 2: Simplify Using the Tangent Value at Specific Angles

We need to recognize that \( \tan 2\pi = 0 \) because the tangent of any multiple of \( \pi \) is zero. Therefore, the expression simplifies as: \( \tan(2\pi - x) = \frac{0 - \tan x}{1 + 0 \cdot \tan x} = -\tan x \).

Step 3: Express the Result

Now that we have simplified the expression using the given trigonometric identity, we express the final answer as a function of \( x \). Thus, \( \tan(2\pi - x) = -\tan x \).

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools in mathematics that simplify complex expressions involving angles and trigonometric functions such as sine, cosine, and tangent. These identities are relationships and equations that hold true for all angles and are used to solve equations, simplify expressions, and prove mathematical relationships. Here are a few basic identities:

• Pythagorean Identity: It states that for any angle \( \theta \), \( \sin^2 \theta + \cos^2 \theta = 1 \).
• Angle Addition and Subtraction: For any angles \( a \) and \( b \):
  \( \sin(a \pm b) = \sin a \cos b \pm \cos a \sin b \),
  \( \cos(a \pm b) = \cos a \cos b \mp \sin a \sin b \),
  \( \tan(a \pm b) = \frac{\tan a \pm \tan b}{1 \mp \tan a \tan b} \).
• Even-Odd Identities: These help to simplify expressions by recognizing the symmetry of trigonometric functions:
  \( \sin(-\theta) = -\sin \theta \),
  \( \cos(-\theta) = \cos \theta \),
  \( \tan(-\theta) = -\tan \theta \).

Using these identities, you can rewrite expressions involving trigonometric functions more simply or in a form that better suits your needs. They also form the foundation for solving more complex mathematical problems.

Tangent Function

The tangent function, denoted as \( \tan(\theta) \), is one of the primary trigonometric functions that describes the ratio of the sine and cosine of an angle. Specifically, \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \). The tangent function is periodic, with a period of \( \pi \), meaning it repeats its values every \( \pi \) radians or 180 degrees.
The function is defined for all angles except where \( \cos(\theta) = 0 \), which occurs at odd multiples of \( \pi/2 \). At these points, the function has vertical asymptotes.

Tangent has some distinctive properties:

• The tangent function is an odd function, implying that \( \tan(-\theta) = -\tan(\theta) \).
• The function is positive in the first and third quadrants and negative in the second and fourth quadrants of the unit circle.

In the context of simplifying the expression \( \tan(2\pi - x) \), knowing that \( \tan(2\pi) = 0 \) is a key piece of information. This allows us to simplify the original expression directly to \( -\tan(x) \), following the properties of the tangent function and its periodicity.

Angle Simplification

Angle simplification is a critical process in trigonometry that involves rewriting an angle in a simpler or more recognizable form. This is especially useful when dealing with trigonometric expressions or identities, as it can lead to significant simplification of the entire expression.

To simplify the expression \( \tan(2\pi - x) \), angle identities and properties of trigonometric functions are used:

• Recognizing specific angle trigonometric values is essential. For instance, \( \tan(2\pi) = 0 \) simplifies our task significantly.
• Utilizing symmetry and periodicity of trigonometric functions assists in transforming complex angles. Here, angles of \( 2\pi - x \) benefit from the periodic nature of the tangent, as \( 2\pi \) corresponds to a full circle rotation, leading back to the starting point.

Such simplifications not only make mathematical expressions more manageable but also pave the way for accurate calculations in both homework exercises and real-world applications like engineering and physics.