Problem 29 - A Graphical Approach to Precalculus with Limits

Question

Use identities to write each expression as a function with \(x\) as the only argument. $$\tan \left(180^{\circ}-x\right)$$

Step-by-Step Solution

Answer

\( \tan(180^\circ - x) = -\tan(x) \)

Step 1: Understand the Identity

The identity we will use here is the tangent subtraction identity: \[ \tan(180^\circ - x) = -\tan(x) \]. This identity comes from the properties of the tangent function and angles.

Step 2: Apply the Identity

Substitute the given expression into the identity: \[ \tan(180^\circ - x) = -\tan(x) \]. This shows that the original expression simplifies directly to \( -\tan(x) \).

Step 3: Rewrite the Expression

Rewrite the expression \( \tan(180^\circ - x) \) using the result from Step 2. The expression simplifies to just \(-\tan(x)\), which is now expressed with \( x \) as the only argument.

Key Concepts

Tangent Function

The tangent function is one of the primary trigonometric functions, along with sine and cosine. In trigonometry, it relates the angles of a right triangle to the ratio of two of its sides: the opposite side and adjacent side. Mathematically, the tangent of an angle θ in a right-angled triangle is defined as: \[\tan(θ) = \frac{\text{opposite}}{\text{adjacent}}\]The tangent function is periodic, repeating its values every 180 degrees or \(\pi\) radians. This periodicity is crucial when working with various trigonometric identities and equations.

• Tangent is undefined for angles where the cosine equals zero, such as 90 degrees (or \(\frac{\pi}{2}\) radians).
• The function produces positive values in the first and third quadrants and negative values in the second and fourth quadrants of the unit circle.

Understanding the properties of the tangent function helps in tackling problems in trigonometry, where simplification of expressions is often required.

Angle Subtraction Identity

The angle subtraction identity is an important trigonometric identity involving the tangent function. It allows us to express the tangent of the difference between two angles in terms of the tangents of the individual angles. For the tangent function, the identity is:\[\tan(A - B) = \frac{\tan(A) - \tan(B)}{1 + \tan(A)\tan(B)}\]However, when the older part of the angle involves 180 degrees (like in the given problem), this identity simplifies significantly due to the properties of the unit circle and the symmetry of trigonometric functions.
In the case of \(\tan(180^\circ - x)\), we utilize the fact that tangent has an odd function property, meaning \(\tan(-x) = -\tan(x)\). Thus:

• \(\tan(180^\circ) = 0\), simplifying the subtraction further.
• This results in \(\tan(180^\circ - x) = -\tan(x)\).

By using this identity, complex expressions can be simplified more easily, helping with problem-solving.

Function Simplification

Simplifying trigonometric functions involves reducing complex expressions into simpler terms, often making problems easier to solve or analyze. When simplifying expressions involving trigonometric identities, it is essential to apply relevant properties or identities directly related to the functions involved.
In the original exercise:

• The expression \(\tan(180^\circ - x)\) was simplified using the subtraction identity to become \(-\tan(x)\).
• By using such identities, we cut down the expression to only involve one trigonometric term with \(x\) as the single variable.

Function simplification is an invaluable skill, especially in calculus, physics, and engineering, where it enables more efficient computation and deeper insight into the behavior of functions. Understanding simplification techniques helps in grasping complex problems with greater ease and clarity.