Problem 16 - 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. $$-\frac{7 \pi}{4}$$

Step-by-Step Solution

Answer

The reference angle for \(-\frac{7 \pi}{4}\) is \(\frac{\pi}{4}\).

Step 1: Convert the angle into positive form

First, add \(2\pi\) to the given angle \(-\frac{7 \pi}{4}\) to make it positive. The formula for converting a negative angle into positive is \(angle + 2\pi\). Hence, the positive angle for \(-\frac{7 \pi}{4}\) is \(-\frac{7 \pi}{4} + 2\pi = -\frac{7 \pi}{4} + \frac{8 \pi}{4}= \frac{\pi}{4}\).

Step 2: Identify the reference angle using the unit circle

Once the angle is in positive form, find the reference angle using the unit circle. The reference angle for any angle in standard position is the acute angle formed by the terminal side of that angle and the x-axis. In other words, it's the smallest angle that the terminal side makes with the x-axis. Since the angle obtained in Step 1, \(\frac{\pi}{4}\) lies in the first quadrant and all angles in first quadrant are same as their reference angles, the reference angle for \(\frac{\pi}{4}\) is \(\frac{\pi}{4}\) itself.

Key Concepts

Unit Circle

The unit circle is a circle with a radius of one unit, centered at the origin of a coordinate plane. Understanding the unit circle is essential for studying trigonometry and angles. All points on the unit circle are of the form \((\cos \theta, \sin \theta)\), where \(\theta\) is the angle formed with the positive x-axis.

• Angles are measured in radians or degrees. In the context of the unit circle, we often use radians.
• The complete circumference of the unit circle corresponds to an angle of \(2\pi\) radians or \(360^\circ\).
• The x-axis acts as the starting line for measuring angles on the unit circle, beginning from \(0\) radians.

By understanding where an angle lies on the unit circle, you can easily determine its sine and cosine values, which are critical for finding reference angles and solving trigonometric equations.

Positive Angle Conversion

Converting negative angles to positive ones is a fundamental skill in trigonometry. It's crucial because it allows us to understand angles in a standardized form, which is easier for calculating reference angles.

To convert a negative angle to a positive one:

• Add \(2\pi\) (or \(360^\circ\)) to the negative angle since these are full rotations on the circle.
• The resulting angle will now be in the same position on the unit circle as the original, but expressed as a positive measure.

For example, given an angle \(-\frac{7\pi}{4}\), adding \(2\pi\) (which is \(\frac{8\pi}{4}\) in terms of common denominators) gives \(\frac{\pi}{4}\). Using this method helps to identify the smallest positive equivalent of an angle, making trigonometric calculations more straightforward.

Quadrants of Angles

Angles on the unit circle are divided into four quadrants, each representing a range of angle measures. Understanding which quadrant an angle resides in helps in finding its reference angle and applying trigonometric identities.

• First Quadrant: Angles between \(0\) and \(\frac{\pi}{2}\) radians (or \(0\) to \(90^\circ\)). Here, all trigonometric functions are positive.
• Second Quadrant: Angles between \(\frac{\pi}{2}\) and \(\pi\) radians (or \(90^\circ\) to \(180^\circ\)). Sine is positive, while cosine and tangent are negative.
• Third Quadrant: Angles between \(\pi\) and \(\frac{3\pi}{2}\) radians (or \(180^\circ\) to \(270^\circ\)). Here, tangent is positive, but sine and cosine are negative.
• Fourth Quadrant: Angles between \(\frac{3\pi}{2}\) and \(2\pi\) radians (or \(270^\circ\) to \(360^\circ\)). Cosine is positive, whereas sine and tangent are negative.

By determining the quadrant of an angle, you can further deduce the sign of its trigonometric values, aiding in accurate finding of reference angles and solutions.