Problem 9 - Precalculus Essentials

Question

In Exercises \(9-16\), evaluate the trigonometric function at the quadrantal angle, or state that the expression is undefined. $$\cos \pi$$

Step-by-Step Solution

Answer

\(\cos(\pi) = -1\)

Step 1: Understand the angle

Convert the given angle from radians to degrees to help visualize it better. Recall that \(\pi\) radians is equal to 180 degrees.

Step 2: Cosine at 180 degrees

Next, consider where 180 degrees, or \(\pi\) radians, is on the unit circle. Since cosine value at this angle corresponds to the x-coordinate on the unit circle, observe that the x-coordinate at this location is -1.

Step 3: Return the Cosine Value

With all this information, it can be concluded that \(\cos(\pi) = -1\).

Key Concepts

Quadrantal Angles

When dealing with trigonometric functions, encountering quadrantal angles is common. Quadrantal angles are angles located on the axes of the coordinate plane. They divide the circle into four quadrants. The quadrantal angles in degrees are 0°, 90°, 180°, and 270°. In radians, these are 0, \( \frac{\pi}{2} \), \( \pi \), and \( \frac{3\pi}{2} \).

• 0° or 0 radians is the positive x-axis.
• 90° or \( \frac{\pi}{2} \) radians is the positive y-axis.
• 180° or \( \pi \) radians is the negative x-axis.
• 270° or \( \frac{3\pi}{2} \) radians is the negative y-axis.

Understanding these angles is crucial because their sine and cosine values are easily determined and often result in simple outputs like -1, 0, or 1. When an angle is a quadrantal angle, its associated trigonometric function takes values on or off the axes, simplifying calculations remarkably.

Unit Circle

The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of 1 centered at the origin of the coordinate plane. The unit circle helps visualize and understand the relationships of trigonometric functions and angles.
On the unit circle:

• The x-coordinates represent cosine values.
• The y-coordinates represent sine values.
• Different angles correspond to different points on the circle.
• For example, at an angle of \( \pi \) or 180°, the point is (-1, 0).

This reveals directly that the cosine value is -1. The negative sign indicates the direction along the x-axis on the left side of the origin. When solving trigonometric problems, the unit circle is immensely helpful, particularly for quadrantal angles because it provides clear x and y values with immediate interpretive insight.

Cosine Function

The cosine function is one of the primary trigonometric functions and is crucial in many areas of mathematics. The function is periodic, meaning it repeats values in a consistent interval, specifically every \( 2\pi \) radians (360°).

• Cosine of any angle corresponds to the x-coordinate on the unit circle.
• At \( \pi \) radians (180°), cosine is -1, indicating full extension along the negative x-axis.
• The cosine function varies from -1 to 1 depending on the angle.

By understanding the cosine function, it's easier to determine the value of cosine for any angle. Its specific relation to the unit circle simplifies calculations, especially for angles like quadrantal ones. Using cosine effectively in problem-solving is a matter of recognizing its relation to triangles and the circle.