Problem 2 - Algebra 2

Question

Use the graph of \(y=\tan \theta\) to find each value. If the tangent is undefined at that point, write undefined. $$ \tan \pi $$

Step-by-Step Solution

Answer

The value of \(\tan \pi\) is 0.

Step 1: Understanding the underline meaning of the tan function

\(\tan \theta\) is a trigonometric function. It is defined as the ratio of the y-coordinate to the x-coordinate of a point on the unit circle. If that point is on the x-axis, the y-coordinate is zero, which means \(\tan \theta\) is zero. This is the case for \(\theta = 0\) and \(\theta = \pi\)

Step 2: Evaluate \(\tan \pi\)

From the unit circle or from the graph of the tan function you can see that \(\tan \pi = 0\).

Key Concepts

Tangent Function

The tangent function, denoted as \( \tan \theta \), is one of the primary trigonometric functions, alongside sine and cosine. It is predominantly used to model periodic phenomena and can be found throughout various fields of science and engineering.
The tangent of an angle \( \theta \) in a right triangle is defined as the ratio of the opposite side to the adjacent side. However, in the context of the unit circle, the tangent function is defined differently:

• \( \tan \theta = \frac{y}{x} \)
• \( y \) is the y-coordinate of the point on the unit circle
• \( x \) is the x-coordinate of that same point

Hence, the tangent function becomes very significant when analyzing angles that rotate around the unit circle.
Additionally, the tangent function is periodic with a period of \( \pi \). This means its values repeat every \( \pi \) radians, which can be easily observed in its graph. Recognizing the repeating nature and its coordinates is key to understanding problems involving the tangent function.

Unit Circle

The unit circle is a circle with a radius of one unit, centered at the origin of a coordinate plane. It is a fundamental tool for understanding trigonometric functions and their properties.
The purpose of the unit circle in trigonometry is to relate the lengths of a right triangle's sides to angles in a simple circle.

• The circle allows for the definition of the sine, cosine, and tangent functions using the coordinates \((x, y)\).
• For any angle \(\theta\), the cosine of the angle is the x-coordinate, and the sine is the y-coordinate of the circle's intersection with the terminal side of the angle.
• The tangent function, as previously noted, can be derived using these coordinates: \(\tan \theta = \frac{y}{x}\).

The beauty of the unit circle is that it simplifies the computation of trigonometric functions for special angles like \(\pi\) radians. Here, the coordinates are \((-1, 0)\), demonstrating that \(\tan \pi = 0\) because the y-coordinate is zero.

Undefined Values

In trigonometry, you may often encounter situations where a function becomes undefined. Knowing when and why a function is undefined is crucial to solving trigonometric problems.
For the tangent function, it becomes undefined when the x-coordinate of the corresponding point on the unit circle is zero. This is because division by zero is mathematically undefined.

• Common situations for tangent's undefined values are angles like \(\frac{\pi}{2}\), \(\frac{3\pi}{2}\), etc.
• These points correspond to the top or the bottom of the unit circle, where any value divided by zero would yield an undefined result.
• Understanding undefined points of the tangent function will help avoid errors when sketching its graph or solving trigonometric equations.

By identifying these values, you gain control over trigonometric functions and avoid pitfalls' complexities, ensuring that you choose accurate approaches when tackling mathematical problems.