Problem 37 - Algebra 2

Question

Sketch each angle. Then find its reference angle. \(\frac{5 \pi}{6}\)

Step-by-Step Solution

Answer

The reference angle of \( \frac{5\pi}{6} \) is \( \frac{\pi}{6} \).

Step 1: Identify the Quadrant

To determine the reference angle, first identify which quadrant the given angle \( \frac{5\pi}{6} \) lies in. Since the angle is greater than \( \frac{\pi}{2} \) and less than \( \pi \), it is in the second quadrant.

Step 2: Understand Reference Angle Concept

The reference angle is the acute angle formed by the terminal side of the given angle with the x-axis. For angles in the second quadrant, the reference angle \( \theta_{ref} \) is \( \pi - \theta \).

Step 3: Calculate the Reference Angle

Using the formula \( \theta_{ref} = \pi - \theta \), substitute \( \theta = \frac{5\pi}{6} \) : \[ \theta_{ref} = \pi - \frac{5\pi}{6} \].

Step 4: Simplify the Expression

Simplify the expression to find the reference angle: \[ \theta_{ref} = \frac{6\pi}{6} - \frac{5\pi}{6} = \frac{\pi}{6} \].

Step 5: Sketch the Angle

Represent the angle \( \frac{5\pi}{6} \) by drawing a diagram in the second quadrant. Start from the positive x-axis, move counterclockwise to form a line with the angle \( \frac{5\pi}{6} \), which is slightly less than \( \pi \). Highlight the reference angle as \( \frac{\pi}{6} \) between the terminal side and the x-axis.

Key Concepts

Angle Quadrants

The concept of angle quadrants is essential when dealing with angles in trigonometry. In the coordinate plane, the axes divide the plane into four distinct sections, known as quadrants. Each quadrant represents a unique range of angle degrees or radians.

• Quadrant I: Here, angles range from 0 to \(\frac{\pi}{2}\) (0 to 90 degrees).
• Quadrant II: Angles range from \(\frac{\pi}{2}\) to \(\pi\) (90 to 180 degrees).
• Quadrant III: Angles range from \(\pi\) to \(\frac{3\pi}{2}\) (180 to 270 degrees).
• Quadrant IV: Angles range from \(\frac{3\pi}{2}\) to \(2\pi\) (270 to 360 degrees).

For the angle \(\frac{5\pi}{6}\), we're in Quadrant II. Knowing the quadrant helps ascertain angle behavior and simplifies calculating the reference angle.

Acute Angles

An acute angle is simply any angle that measures less than \(\frac{\pi}{2}\) or 90 degrees. These angles are small and sharp. In the context of reference angles, acute angles are particularly important.

Why? Because reference angles are always acute! They serve as the smallest angle between the terminal side of a given angle and the x-axis.

For our exercise with \(\frac{5\pi}{6}\), the reference angle is \(\frac{\pi}{6}\), which is acute since it measures less than 90 degrees.

Terminal Side

The terminal side of an angle is one of the two sides that form the angle. The initial side usually lies on the positive x-axis, and from there, the terminal side is rotated to form the complete angle.

In our exercise, the terminal side of \(\frac{5\pi}{6}\) concludes in Quadrant II. It's the terminal side that's used to calculate the reference angle. Formulated by the acute angle created between this terminal side and the x-axis.

Representation on a coordinate plane allows visualization of the terminal side, making comprehension of the angle and reference angle much clearer.

X-axis

The x-axis is a staple in the Cartesian coordinate system. It's the horizontal line that spans across the plane. In trigonometry, the x-axis is crucial as it serves as the starting point for measuring angles.

When finding reference angles, we measure the smallest angle made between an angle's terminal side and the x-axis. This is true regardless of which quadrant the angle terminates in.

For instance, in our example with \(\frac{5\pi}{6}\), the reference angle \(\frac{\pi}{6}\) is found between the terminal side of the angle and the x-axis. Highlighting the x-axis helps in visually determining reference angles efficiently when sketching angles.