Problem 8 - Algebra and Trigonometry

Question

Find the reference angle for the given angle. $$ \begin{array}{llll}{\text { (a) } \frac{4 \pi}{3}} & {\text { (b) } \frac{33 \pi}{4}} & {\text { (c) }-\frac{23 \pi}{6}}\end{array} $$

Step-by-Step Solution

Answer

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

Step 1: Understanding Reference Angle Concept

Reference angles are the acute angles formed with the horizontal axis. They are always positive and between \(0\) and \( \frac{\pi}{2} \) radians. The reference angle can be found for any given angle by determining how far it is from the nearest x-axis (either \(0\), \(\pi\), or \(2\pi\)).

Step 2: Reference Angle for \(\frac{4\pi}{3}\)

The angle \(\frac{4\pi}{3}\) is in the third quadrant since \(\pi < \frac{4\pi}{3} < \frac{3\pi}{2}\). The formula for finding the reference angle in the third quadrant is: \( \text{Reference Angle} = \theta - \pi \). Substituting \(\theta = \frac{4\pi}{3}\), the calculation is \(\frac{4\pi}{3} - \pi = \frac{4\pi}{3} - \frac{3\pi}{3} = \frac{\pi}{3}\).

Step 3: Reference Angle for \(\frac{33\pi}{4}\)

First, convert \(\frac{33\pi}{4}\) to its equivalent within \([0, 2\pi]\) by subtracting multiples of \(2\pi\). Calculate \(\frac{33\pi}{4} - 8\pi = \frac{33\pi}{4} - \frac{32\pi}{4} = \frac{\pi}{4}\). The reference angle for \(\frac{\pi}{4}\) is \(\frac{\pi}{4}\) itself as it is already an acute angle.

Step 4: Reference Angle for \(-\frac{23\pi}{6}\)

Convert \(-\frac{23\pi}{6}\) to a positive angle by adding multiples of \(2\pi\) until the angle is positive. Compute \(-\frac{23\pi}{6} + 4\pi = -\frac{23\pi}{6} + \frac{24\pi}{6} = \frac{\pi}{6}\). Since \(\frac{\pi}{6}\) is an acute angle, the reference angle is \(\frac{\pi}{6}\).

Key Concepts

Trigonometric Angles

Trigonometric angles are fundamental in understanding many areas of mathematics, especially in trigonometry. They describe the size of the angle in a circle and are often expressed in degrees or radians. Trigonometric angles help in defining the position of a point on the unit circle, which is crucial for understanding trigonometric functions.

Key things to know about trigonometric angles:

• Measured from the positive x-axis, extending counter-clockwise.
• Angles in different quadrants affect the signs of trigonometric functions (sine, cosine, tangent).
• Standard angle positions:
  • First quadrant: Between 0 and 90 degrees (or 0 and \(\frac{\pi}{2}\) radians)
  • Second quadrant: Between 90 and 180 degrees (or \(\frac{\pi}{2}\) and \(\pi\) radians)
  • Third quadrant: Between 180 and 270 degrees (or \(\pi\) and \(\frac{3\pi}{2}\) radians)
  • Fourth quadrant: Between 270 and 360 degrees (or \(\frac{3\pi}{2}\) and \(2\pi\) radians)

The above points will help when using angles to solve problems such as finding reference angles or using trigonometric identities.

Radian Measure

Radian measure is a way to express angles based on the radius of a circle. Unlike degrees, which divide a circle into 360 equal parts, radians relate the arc length directly to the radius. This reveals a natural and mathematically convenient way to measure angles.

Things to know about radians:

• One complete revolution around a circle equals \(2\pi\) radians.
• \(\pi\) radians equal 180 degrees, which is a semi-circle.
• Radians simplify many mathematical formulas in calculus and physics.
• To convert degrees to radians, use the formula \(\text{radians} = \text{degrees} \times \frac{\pi}{180}\).
• To convert radians to degrees, use \(\text{degrees} = \text{radians} \times \frac{180}{\pi}\).

Understanding radians is crucial when working with trigonometric functions, as these functions often assume input in radians in calculus and higher-level math.

Acute Angles

Acute angles are those angles that are less than 90 degrees or \(\frac{\pi}{2}\) radians. These angles play a special role in trigonometry as they are often used to find reference angles.

Properties of acute angles include:

• Always less than a right angle.
• Non-negative and less than \(\frac{\pi}{2}\) radians.
• When an angle is expressed as an acute angle, it can simplify calculations involving triangle properties and trigonometric identities.
• Reference angles are always converted to acute angles to simplify the understanding of trigonometric functions across different quadrants.

Having a good grasp of acute angles will make it easier to determine reference angles and solve trigonometric problems efficiently.

Unit Circle

The unit circle is a essential concept in trigonometry. It is a circle with a radius of 1 unit, centered at the origin (0,0) of the coordinate plane. The unit circle aids in defining the sine, cosine, and tangent functions for any angle.

Important features of the unit circle:

• Any point on the unit circle can be described by the coordinates \((\cos(\theta), \sin(\theta))\), where \(\theta\) is the angle formed at the origin with the positive x-axis.
• The circle’s circumference is \(2\pi\), which corresponds to a full revolution or \(360\) degrees.
• Understanding positions on the unit circle is especially useful for analyzing angles beyond 0 to \(\frac{\pi}{2}\), including those in all four quadrants.
• It's vital for solving trigonometric equations, understanding periodic functions, and working with complex numbers.
• Tangents are lines that touch the circle at exactly one point, with slopes given by \(\tan(\theta)\).

Using the unit circle allows students to visualize and compute the values of trigonometric functions, not only for basic angles but for a vast range of angles including their supplementary, complementary, and negative counterparts.