Problem 30 - A Graphical Approach to Precalculus with Limits

Question

Use identities to write each expression as a function with \(x\) as the only argument. $$\tan \left(360^{\circ}-x\right)$$

Step-by-Step Solution

Answer

\(-\tan(x)\)

Step 1: Understand the Trigonometric Identity

We need to use the identity for the tangent of a difference: \( \tan(360^\circ - x) \). We know that \( \tan(360^\circ - x) = -\tan(x) \), based on the periodicity and properties of tangent. Specifically, tangent of angles in the fourth quadrant can be converted to a negative tangent of the equivalent angle in the first quadrant.

Step 2: Apply the Identity to Simplify

Using the identity, substitute into the expression: \( \tan(360^\circ - x) = -\tan(x) \). This simplifies the expression to a single function of \( x \), which is \( -\tan(x) \).

Key Concepts

Tangent Function

The tangent function, often represented by \( \tan(x) \), is one of the primary trigonometric functions, alongside sine and cosine. It is defined as the ratio of the sine of an angle to the cosine of that angle. That is, \( \tan(x) = \frac{\sin(x)}{\cos(x)} \). This function is particularly useful because it relates the vertical and horizontal components of an angle in a right triangle.

The key characteristic of the tangent function is that it can experience drastic changes in value since it involves division. Whenever the cosine of an angle approaches zero, the tangent function can go to infinity, due to division by a very small number.

• It is undefined for angles where \( \cos(x) = 0 \), which happens at \( x = \frac{\pi}{2} + k\pi \) for any integer \( k \).
• The tangent function's graph has a series of repeating vertical asymptotes at these points.
• This function repeats every \( \pi \) radians, making it periodic.

Periodicity of Trigonometric Functions

Periodicity is a fundamental concept in understanding trigonometric functions like tangent, sine, and cosine. These functions repeat their values in regular intervals, known as periods. The concept of periodicity is crucial because it allows the simplification of expressions involving angles outside the usual first cycle from \( 0 \) to \( 360^{\circ} \) or \( 0 \) to \( 2\pi \) radians.

For the tangent function, the period is \( 180^{\circ} \) or \( \pi \) radians. This means that \( \tan(x) = \tan(x + n\pi) \) for any integer \( n \). This periodicity characteristic is particularly handy when dealing with angles that are modified within the tangent function, such as \( 360^{\circ} - x \).

• Tangent function has its peaks and valleys spaced evenly every \( \pi \) radians.
• This repetitive nature is why transformations like \( \tan(360^{\circ} - x) = -\tan(x) \) hold true.
• It helps in translating angles to a simpler, more conventional form, simplifying many mathematical problems.

Angle Conversion

Converting angles can be a helpful strategy when simplifying trigonometric expressions. Angles in trigonometry can be measured in degrees or radians, and sometimes it is useful to change the way we represent these angles to make calculations easier or recognize patterns.

For example, knowing the relationship between degrees and radians allows us to see that \( 360^{\circ} = 2\pi \) radians. This conversion is crucial in recognizing periodicity and simplifying expressions.

• To convert degrees to radians, multiply by \( \frac{\pi}{180} \).
• To convert radians to degrees, multiply by \( \frac{180}{\pi} \).
• Working with the standard forms of angles helps in applying identities like \( \tan(360^{\circ} - x) = -\tan(x) \), where the angle conversion shows the angle's position in the trigonometric cycle.