Problem 5 - Algebra and Trigonometry

Question

1-8. Find the reference angle for the given angle. (a) \(\frac{11 \pi}{4} \quad\) (b) \(-\frac{11 \pi}{6} \quad\) (c) \(\frac{11 \pi}{3}\)

Step-by-Step Solution

Answer

(a) \(\frac{\pi}{4}\), (b) \(\frac{\pi}{6}\), (c) \(\frac{\pi}{3}\).

Step 1: Understanding Reference Angle

The reference angle is the acute angle that a given angle makes with the horizontal axis (x-axis). It represents the angle's simplest form within a normal 0 to \(90^\circ\) or \(0\) to \(\frac{\pi}{2}\) measured counterclockwise. For angles given in radians, it is beneficial to first understand their relative position within the unit circle.

Step 2: Convert to Equivalent Angle in Unit Circle for (a)

For (a), given \(\frac{11 \pi}{4}\), first convert it to a principal angle by subtracting \(2 \pi\) until it is between \(0\) and \(2\pi\). \[\frac{11 \pi}{4} - 2\pi = \frac{11 \pi}{4} - \frac{8 \pi}{4} = \frac{3 \pi}{4}\]. Since \(\frac{3 \pi}{4}\) is in the second quadrant, the reference angle is \(\pi - \frac{3 \pi}{4} = \frac{\pi}{4}\).

Step 3: Find Principal Angle Equivalent for (b)

For (b) \(-\frac{11 \pi}{6}\), convert to a positive angle by adding \(2 \pi\) until it lies between \(0\) and \(2 \pi\). \[-\frac{11 \pi}{6} + 2 \pi = -\frac{11 \pi}{6} + \frac{12 \pi}{6} = \frac{\pi}{6}\]. \(\frac{\pi}{6}\) already places this in the first quadrant, where it is itself the reference angle.

Step 4: Normalize the Angle for (c)

For (c) \(\frac{11 \pi}{3}\), subtract \(2 \pi\) to find the equivalent angle between \(0\) and \(2 \pi\). \[\frac{11 \pi}{3} - 2 \pi = \frac{11 \pi}{3} - \frac{6 \pi}{3} = \frac{5 \pi}{3}\]. The angle \(\frac{5 \pi}{3}\) is in the fourth quadrant, so the reference angle is \(2 \pi - \frac{5 \pi}{3} = \frac{\pi}{3}\).

Key Concepts

Understanding the Unit Circle

The unit circle is a fundamental concept in trigonometry. It's a circle with a radius of 1, centered at the origin of the coordinate plane. This simple geometric tool serves as the basis for defining trigonometric functions for all real numbers. Here's how it works:

• The circle's equation is given by \(x^2 + y^2 = 1\).
• As you move around the circle, the angle in radians from the positive x-axis defines the position. These angles can be positive, if measured counterclockwise, and negative if measured clockwise.

Using the unit circle, one can see where angles fall, helping to convert them into their principal and reference angles. For our exercise, calculating the reference angle requires knowing which quadrant an angle terminates in on the unit circle. Each quadrant has unique rules about reference angles to ensure they remain acute.

Identifying Principal Angles

Principal angles are angles that fall within one rotation of the unit circle, specifically between 0 and \(2\pi\) radians (or 0 and \(360^\circ\)). Finding the principal angle involves normalizing an angle by adding or subtracting \(2\pi\) until it fits within this range.

• For modal rotations like those examined in the original exercise, the transformation aims to simplify complex angles.
• The principal angle helps identify the corresponding quadrant in which the reference angle is computed.

Let's illustrate with an example: if you have an angle like \(\frac{11\pi}{4}\), you must subtract \(2\pi\) (or \(\frac{8\pi}{4}\)) to normalize it to \(\frac{3\pi}{4}\), identifying its placement in the second quadrant. Understanding this concept is instrumental in analyzing angles and solving trigonometric problems.

Exploring Trigonometric Functions

Trigonometric functions—sine, cosine, and tangent—are derived from the unit circle and necessitate a thorough understanding of angles and their reference counterparts. These functions allow us to relate the angles to side lengths in right triangles, and correspond to specific coordinates of points on the unit circle.

• Sine is associated with the y-coordinate.
• Cosine corresponds to the x-coordinate.
• Tangent is the ratio of sine to cosine.

Given an angle's location on the unit circle, you can quickly determine these function values by understanding the angle's reference and principal forms. Whether an angle is in the first quadrant or somewhere in the third, knowing the quadrant helps predict the sign and magnitude of these functions, crucial for problem-solving in trigonometry.