Problem 104 - Precalculus

Question

For the following exercises, state the reference angle for the given angle. $$ \frac{5 \pi}{6} $$

Step-by-Step Solution

Answer

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

Step 1: Identify the quadrant

The angle \( \frac{5\pi}{6} \) is in the second quadrant because \( \pi = \frac{6\pi}{6} \) and \( \frac{5\pi}{6} \) is less than \( \pi \). Angles in the second quadrant are between \( \frac{\pi}{2} \) and \( \pi \).

Step 2: Use reference angle formula for the second quadrant

For angles in the second quadrant, the reference angle \( \theta' \) can be found using the formula \( \theta' = \pi - \theta \). In this case, substitute \( \theta = \frac{5\pi}{6} \).

Step 3: Calculate the reference angle

Substitute the given angle into the reference angle formula: \[\theta' = \pi - \frac{5\pi}{6} = \frac{6\pi}{6} - \frac{5\pi}{6} = \frac{\pi}{6}\]So the reference angle is \( \frac{\pi}{6} \).

Key Concepts

Trigonometric Quadrants

Understanding trigonometric quadrants is essential when dealing with angles and their reference angles. The coordinate plane is divided into four quadrants:

• The **first quadrant** where both x and y coordinates are positive.
• The **second quadrant** where x is negative and y is positive.
• The **third quadrant** where both x and y are negative.
• The **fourth quadrant** where x is positive and y is negative.

Each quadrant contains angles from a full circle, which is 360 degrees or \(2\pi\) radians. Angles in the quadrants help us determine the signs of sine, cosine, and tangent for those angles. For example, sine is positive in the first and second quadrants, as y is positive there. Recognizing which quadrant an angle lies in helps simplify solving problems involving angles and trigonometric functions.

Second Quadrant

In the second quadrant, angles range from \(\frac{\pi}{2}\) to \(\pi\) radians. This range is equivalent to 90 to 180 degrees. When working with angles in the second quadrant, **sine** values remain positive, while **cosine** and **tangent** values are negative. This is because, in this quadrant, the x-values on the coordinate plane are negative, reflecting negatively on cosine and tangent functions.
To determine if an angle is in the second quadrant, ensure it is greater than \(\frac{\pi}{2}\) yet less than \(\pi\). For instance, an angle like \(\frac{5\pi}{6}\) fits perfectly in this quadrant because \(\frac{\pi}{2} = \frac{3\pi}{6}\) and \(\pi = \frac{6\pi}{6}\). Notice that this quadrant boundary helps quickly identify the sign of trigonometric functions for various angles within its range.

Reference Angle Formula

A reference angle is the smallest angle that the terminal side of the given angle makes with the x-axis. Calculating the reference angle is crucial for simplifying trigonometric functions, especially since these values are often the ones you'll use for solving practical problems.
For angles in the second quadrant, the reference angle \(\theta'\) is found using the formula:\[\theta' = \pi - \theta\]This formula emphasizes subtracting the given angle from \(\pi\), confirming what remains is the acute angle (less than 90 degrees) needed.
For example, if you have an angle \(\frac{5\pi}{6}\), use the formula to get:\[\theta' = \pi - \frac{5\pi}{6} = \frac{6\pi}{6} - \frac{5\pi}{6} = \frac{\pi}{6}\]The result, \(\frac{\pi}{6}\), is the reference angle. This technique helps provide a consistent and straightforward way of determining angles needed for further trigonometric analysis.