Problem 13 - Precalculus Essentials

Question

In Exercises \(9-16\), evaluate the trigonometric function at the quadrantal angle, or state that the expression is undefined. $$\tan \frac{3 \pi}{2}$$

Step-by-Step Solution

Answer

The tangent of \(\frac{3 \pi}{2}\) is undefined.

Step 1: Identify the quadrantal angle

We have the quadrantal angle as \(\frac{3 \pi}{2}\), which corresponds to the point on the unit circle where x = 0 and y = -1.

Step 2: Evaluate the tangent function

The tangent function is defined as the ratio of the y-coordinate to the x-coordinate. For the angle \(\frac{3 \pi}{2}\), the x-coordinate of the corresponding point on the unit circle is 0, and the y-coordinate is -1. So, \(\tan \frac{3 \pi}{2} = \frac{-1}{0}\).

Step 3: Define the result of the tangent function

The expression \(\frac{-1}{0}\) is undefined because division of any number by zero is undefined in mathematical expression.

Key Concepts

Quadrantal Angles

Quadrantal angles are special angles at which the terminal side lies on the axes of the unit circle. These angles are integral multiples of \(\pi/2\) (or 90°). Quadrantal angles are represented at 0, \(\pi/2\), \(\pi\), \(3\pi/2\), and \(2\pi\).

They are important because the trigonometric functions often have distinct values or properties at these points.

• At \(0\), the point on the unit circle is (1,0).
• At \(\pi/2\), the point is (0,1).
• At \(\pi\), the point is (-1,0).
• At \(3\pi/2\), the point is (0,-1).
• At \(2\pi\), the point cycles back to (1,0).

Understanding these angles helps evaluate trigonometric functions easily since you'll know the exact coordinates that these angles point to on the unit circle.

Tangent Function

The tangent function, denoted as \( \tan \theta\), relates the angle to the slope of the corresponding line in the coordinate plane. It is calculated as the ratio of the y-coordinate to the x-coordinate in the unit circle, or simply\[\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{y}{x}.\]

This means that the tangent of an angle tells us how steep the line is. For angles like \(3\pi/2\), where x is 0, the tangent function becomes undefined.
- If the x-coordinate is zero, dividing by it results in an undefined slope akin to having a vertical line.- Thé significance of an undefined tangent hints at the steepness being infinite or nonexistent, depending on the approach.
In practical scenarios, undefined tangent values indicate critical boundaries for graphing and calculations involving slopes or rotations.

Unit Circle

The unit circle is a perfect circle with a radius of 1, centered at the origin (0,0) of a coordinate plane. It provides a geometric way of understanding trigonometric functions since all the trig functions can be derived from the coordinates of points on this circle.

Key features of the unit circle:

• Each point on the circle can be associated with an angle \(\theta\) that originated from the positive side of the x-axis.
• The x-coordinate of a point on the unit circle is \(\cos \theta\) and the y-coordinate is \(\sin \theta\).
• Quadrantal angles, such as \(3\pi/2\), correspond to points where either the x or y coordinate is 0.

The unit circle is fundamentally crucial because it allows for a consistent way to visualize and calculate the values of sine, cosine, and tangent functions, especially for common angles like quadrantal angles. Understanding this circle makes it easy to predict the behavior of these functions as angles change.