Problem 49 - Precalculus: A Concise Course

Question

Find the reference angle \(\boldsymbol{\theta}^{\prime}\), and sketch \(\boldsymbol{\theta}\) and \(\boldsymbol{\theta}^{\prime}\) in standard position. $$ \theta=\frac{2 \pi}{3} $$

Step-by-Step Solution

Answer

The reference angle \( \theta' = \frac{\pi}{3} \). Both angles are sketched starting from the positive x-axis, with \( \theta \) extending into the second quadrant and \( \theta' \) residing in the first quadrant.

Step 1: Identify the Quadrant

First, understand the quadrant in which \( \theta \) lies. Since \( \theta = \frac{2 \pi}{3} \) is between \( \pi/2 \) and \( \pi \), it's established that \( \theta \) lies in the second quadrant.

Step 2: Determine the Reference Angle

To find \( \theta' \), the reference angle, subtract from \( \pi \). This formula changes depending on the quadrant the angle is in. For angles in the second quadrant, the formula is \( \pi - \theta \). As such, \( \theta' = \pi - \theta = \pi - \frac{2\pi}{3} = \frac{\pi}{3} \).

Step 3: Sketching the Angles

To sketch \( \theta \) and \( \theta' \), start from the positive x-axis (0 radians), then sketch angles counterclockwise. \( \theta = \frac{2\pi}{3} \) goes into the second quadrant, while \( \theta' = \frac{\pi}{3} \) stops in the first quadrant.

Key Concepts

Quadrant Identification

Identifying the quadrant of an angle is important because it helps us understand certain properties of the angle, like its reference angle and the sign of its trigonometric functions. When working with the unit circle, angles are measured in radians, starting from the positive x-axis and moving counterclockwise.

Quadrants refer to the four sections of the Cartesian coordinate system:

• The first quadrant (QI) spans from 0 to \(\frac{\pi}{2}\).
• The second quadrant (QII) covers from \(\frac{\pi}{2}\) to \(\pi\).
• The third quadrant (QIII) is from \(\pi\) to \(\frac{3\pi}{2}\).
• The fourth quadrant (QIV) runs from \(\frac{3\pi}{2}\) back to \(2\pi\).

To find out where our angle, \(\theta = \frac{2\pi}{3}\), sits, we see it fits in the range for QII because it's between \(\frac{\pi}{2}\) and \(\pi\). Therefore, understanding quadrants helps in computing the reference angle and anticipating the signs of trigonometric values.

Angle Sketching

Sketching angles in standard position on the coordinate plane helps visualize their location and relation to the axes. To sketch an angle, you begin at the positive x-axis, known as the initial side.

From there, angle measurement proceeds counterclockwise:

• First, draw the initial side along the positive x-axis.
• Then, rotate counterclockwise to the terminal side where the angle ends.

For \(\theta = \frac{2\pi}{3}\), start from the x-axis and rotate counterclockwise through the first quadrant and stop in the second quadrant. This is because it equals \(\frac{2\pi}{3} \, radians\), just after \(\frac{\pi}{2} \, radians\), but before \(\pi\) radians.

When sketching the reference angle \(\theta' = \frac{\pi}{3}\), start again from the positive x-axis, but this time stop sooner, once you reach the extent of the first quadrant.

Trigonometric Functions

Trigonometric functions relate angles to ratios of sides in a right triangle. These functions vary based on the quadrant in which an angle lies. Understanding these can predict the behavior of trigonometric functions for any angle.

Let's look at the signs of sine, cosine, and tangent in the different quadrants:

• In the first quadrant, all functions (sine, cosine, tangent) are positive.
• In the second quadrant, sine is positive, while cosine and tangent are negative.
• In the third quadrant, tangent is positive, but sine and cosine are negative.
• Lastly, in the fourth quadrant, cosine is positive, whereas sine and tangent are negative.

For \(\theta = \frac{2\pi}{3}\), located in the second quadrant, we know that its sine value will be positive, while cosine and tangent values will be negative.

When considering the reference angle, \(\theta' = \frac{\pi}{3}\), it becomes crucial, especially in simplifying calculations because it retains the magnitude without regard to the quadrant-induced sign changes.