Problem 10 - Contemporary Precalculus

Question

In Exercises \(1-10,\) use the definition (not a calculator) to find the function value. $$\tan (-13 \pi)$$

Step-by-Step Solution

Answer

Answer: \(\tan(-13\pi) = 0\)

Step 1: Recall the periodic property of the tangent function

The tangent function is periodic with a period of \(\pi\). It means that for any angle \(\theta\), $$\tan(\theta + n\pi) = \tan(\theta)$$ where \(n\) is an integer.

Step 2: Simplify the given angle using the periodic property of tangent

The given angle in the tangent function is \(-13\pi\). Apply the periodic property of the tangent function to simplify the angle: $$\tan(-13\pi) = \tan(-13\pi + n\pi)$$ where \(n\) is an integer.

Step 3: Determine an appropriate integer value for n

We need to find an integer value for \(n\) such that the angle \((-13\pi + n\pi)\) lies within a known tangent range (e.g., between \(0\) and \(\pi\) or between \(-\pi\) and \(0\)). For this case, let \(n=13\). Then, $$-13\pi + 13\pi = 0$$

Step 4: Find the tangent function value

Substitute the simplified angle back into the tangent function, and use the definition of the tangent function to find the value: $$\tan(-13\pi) = \tan(0) = 0$$ So, \(\tan(-13\pi) = 0\).

Key Concepts

Tangent Function

The tangent function is one of the fundamental trigonometric functions alongside sine and cosine. It is defined as the ratio of the sine function to the cosine function. In mathematical terms, for any angle \( \theta \), the tangent is expressed as:\[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \]This function is notable for its unique properties and behavior in trigonometry.
Unlike sine and cosine, which have a range of values between -1 and 1, the tangent function can take any real number value. It evaluates the slope of the line formed by an angle on the unit circle.

• When the cosine of an angle is zero, the tangent function becomes undefined because division by zero is undefined in mathematics.
• The points where tangent is undefined occur at \( \theta = \frac{\pi}{2} + n\pi \), where \( n \) is any integer.

Periodic Property

Trigonometric functions, including the tangent function, are periodic. This means they repeat their values in regular intervals. For the tangent function, this interval, or period, is \( \pi \).
What does this mean?

• The tangent of an angle \( \theta \) has the same value as the tangent of any angle \( \theta + n\pi \), where \( n \) is an integer.
• This periodic property is essential for simplifying calculations of angles that involve large multiples of \( \pi \).
• In simpler terms, shifting the angle by any integer multiple of \( \pi \) will not change the tangent value.

This periodic nature helps in solving problems where we need to reduce complex angles into a more manageable range, such as between 0 and \( \pi \), which is often easier to work with.

Angle Simplification

When working with trigonometric functions, especially in an academic setting, simplifying angles is key for finding function values more easily. Simplification of angles often involves using the periodic properties of the functions. In our example, we are considering the angle \(-13\pi\).
The objective is to simplify the angle using the tangent’s periodicity:

• By choosing \( n = 13 \), we adjust the angle to \( -13\pi + 13\pi = 0 \).
• Substituting back into the tangent function, \( \tan(-13\pi) = \tan(0) \).
• The tangent of 0 is known to be 0, thus simplifying our calculations significantly.

Through simplifying the angle, we can work with more basic angles where the function values are known or straightforward to calculate. This technique is especially useful in mathematics for avoiding complex computations.