Problem 12 - Precalculus : Building Concepts and Connections

Question

Skills This set of exercises will reinforce the skills illustrated in this section. In Exercises \(9-22,\) find the reference angle for each of the angles given. $$-\frac{5 \pi}{4}$$

Step-by-Step Solution

Answer

The reference angle for \(-\frac{5\pi}{4}\) is \(\frac{\pi}{4}\).

Step 1: Understanding Reference Angles

A reference angle is the positive acute angle that can represent an angle of any measure. The reference angle is the smallest angle that the given angle makes with the x-axis. \(-\frac{5\pi}{4}\) is an angle in standard position (origin at (0,0), initial side along the x-axis), and the terminal side of this angle is in the third quadrant.

Step 2: Converting Negative Angle to Positive

The angle given is a negative angle. The general formula for converting a negative angle to positive is \(2\pi + (negative \ angle)\). So, the positive equivalent of \(-\frac{5\pi}{4}\) will be \(2\pi + \left(- \frac{5 \pi}{4} \right) = \frac{3\pi}{4}\). This places the angle in the second quadrant.

Step 3: Finding the Reference Angle

Since the positive angle is found to lie in the second quadrant, its reference angle can be obtained by subtracting it from \(\pi\) (or 180°). Hence, the reference angle of \(-\frac{5\pi}{4}\) is found by \(\pi - \frac{3\pi}{4} = \frac{\pi}{4}\).

Key Concepts

Reference Angles

A reference angle is a useful concept in trigonometry. It helps in simplifying the process of calculating trigonometric functions for any given angle on the coordinate plane. Essentially, a reference angle is always the smallest angle formed between the terminal side of the given angle and the x-axis.

So, why is the reference angle important? Here are some key reasons:

• It makes calculations easier by converting the angle into an acute angle.
• Trigonometric values of an angle and its reference angle have the same absolute values.
• Understanding reference angles can deepen comprehension of trigonometric identities and functions.

This concept plays a crucial role when working with angles beyond the typical 0° to 90° range, especially those in different quadrants.

Negative Angles

Negative angles are angles that rotate clockwise around the origin. This is the opposite of the more common counterclockwise rotation that produces positive angles. Handling negative angles is vital for understanding real-world phenomena where rotations or measurements can happen in any direction.

In order to deal with a negative angle effectively, you can convert it to a positive angle using the formula: \(2\pi + (\text{negative angle})\). By doing this conversion, trigonometric operations become more intuitive because you're working with familiar, positive measures.

When you convert the negative angle \(-\frac{5\pi}{4}\) into a positive angle, you end up with \(\frac{3\pi}{4}\). This step ensures that the calculations remain consistent, regardless of whether an angle is negative or positive.

Quadrants

The Cartesian plane is divided into four regions called quadrants. These quadrants help in determining the sign of trigonometric functions based on the position of the angle. Understanding these can greatly aid in solving trigonometric problems.

Here are some facts about the quadrants:

• Quadrant I: Both x and y coordinates are positive. Angles from 0° to 90° lie here.
• Quadrant II: x is negative and y is positive. Contains angles from 90° to 180°.
• Quadrant III: Both x and y are negative. Angles from 180° to 270° fall into this quadrant.
• Quadrant IV: x is positive and y is negative. Houses angles from 270° to 360°.

Each quadrant has unique properties that determine the nature of angles and reference angles within it. For example, when transforming \(-\frac{5\pi}{4}\), its positive version \(\frac{3\pi}{4}\) lands in the second quadrant, telling you that the reference angle is the supplement to \(\pi\). By knowing the quadrant, you quickly find that the reference angle of \(\frac{3\pi}{4}\) is \(\frac{\pi}{4}\). This method simplifies solving many trigonometric problems.