Problem 56 - Algebra and Trigonometry

Question

find the reference angle for each angle. $$ \frac{17 \pi}{3} $$

Step-by-Step Solution

Answer

The reference angle for \(\frac{17 \pi}{3}\) is \(\frac{\pi}{3}\).

Step 1: Reduce the Given Angle

Firstly, reduce the given angle \(\frac{17 \pi}{3}\) to an angle between \(0\) and \(2\pi\). We know that \(2\pi\) is a full circle, so subtract or add multiples of \(2\pi\) until the angle is within this range. Here we subtract \(2\pi\) multiples of 5 from \(\frac{17 \pi}{3}\) to get it within the \(0\) to \(2\pi\) range. We get \(\frac{17 \pi}{3} - 5 \times 2\pi = \frac{2 \pi}{3}\)

Step 2: Find the Reference Angle

Now, it depends upon which quadrant the angle \(\frac{2 \pi}{3}\) falls in. In this case, it falls in the second quadrant, and we use \(\pi - \text{angle}\) to calculate the reference angle. Therefore, the reference angle is \(\pi - \frac{2 \pi}{3} = \frac{\pi}{3}\)

Key Concepts

Understanding Angle Reduction

When working with angles, especially in trigonometry, it's essential to reduce larger angles to simpler forms. Consider the angle \(\frac{17 \pi}{3}\). This angle exceeds a full circle, which is \(2\pi\) radians, so it's necessary to bring it within the standard range of \(0\) and \(2\pi\).

To achieve this, we use the method of angle reduction:

• Subtract or add multiples of \(2\pi\) until the angle falls within a single circle.
• In our example, \(\frac{17 \pi}{3} - 5 \times 2\pi = \frac{2 \pi}{3}\).

Now the angle \(\frac{2 \pi}{3}\) is between \(0\) and \(2\pi\), making it much easier to work with.

Exploring Quadrants in Trigonometry

Trigonometry divides a circle into four quadrants, each playing a unique role in evaluating angles and their reference angles. After reducing \(\frac{17 \pi}{3}\) to \(\frac{2 \pi}{3}\), we need to determine its position in the circle.

Here's a quick overview of the quadrants:

• First Quadrant: \(0\) to \(\frac{\pi}{2}\)
• Second Quadrant: \(\frac{\pi}{2}\) to \(\pi\)
• Third Quadrant: \(\pi\) to \(\frac{3\pi}{2}\)
• Fourth Quadrant: \(\frac{3\pi}{2}\) to \(2\pi\)

Since \(\frac{2 \pi}{3}\) lies between \(\frac{\pi}{2}\) and \(\pi\), it is in the second quadrant. Identifying the quadrant is crucial for the next step: finding the reference angle.

Calculating the Trigonometric Reference Angle

The reference angle is a small, positive angle that forms a crucial link between the given angle and the x-axis. It simplifies calculations in trigonometry.

When an angle is found in the second quadrant, like \(\frac{2 \pi}{3}\), the formula for calculating the reference angle is \(\pi - \text{angle}\).

Here's how to apply it:

• For the angle \(\frac{2 \pi}{3}\), the reference angle is \(\pi - \frac{2 \pi}{3}\).
• This results in a reference angle of \(\frac{\pi}{3}\).

This reference angle is always positive and less than \(\frac{\pi}{2}\), making trigonometric functions easier to evaluate.