Problem 11 - Precalculus : Building Concepts and Connections

Question

Skills This set of exercises will reinforce the skills illustrated in this section. In Exercises \(9-22,\) find the reference angle for each of the angles given. $$ -\frac{7 \pi}{6} $$

Step-by-Step Solution

Answer

The reference angle for \(-\frac{7 \pi}{6}\) is \(\frac{5 \pi}{6}\).

Step 1: Classify the quadrant

Start by identifying the quadrant the angle falls into when standard position is considered (where the vertex is at the origin and one ray is on the positive x-axis). Angle \(-\frac{7 \pi}{6}\) measures in radians counterclockwise from the positive x-axis. Therefore, it's a negative angle and measured clockwise, placing it in the fourth quadrant.

Step 2: Calculate the reference angle

The reference angle is the acute angle an angle makes with the x-axis. In the fourth quadrant, the formula to find the reference angle is the absolute difference between the angle and \(2\pi\), or in other words, \( |\text{angle} - 2\pi|\). In this case, it would be \( |- \frac{7 \pi}{6} - 2\pi|\). Because \(2\pi\) is equivalent to \( \frac{12 \pi}{6}\), the reference angle becomes: \( |\frac{-7 \pi}{6} - \frac{12 \pi}{6}| = |\frac{-19 \pi}{6}|\). However, we are looking for the smallest positive value, hence, simply add \(2\pi\) until you get a positive value. Thus the reference angle is \( |\frac{-19 \pi}{6} + 2(2\pi)| = |\frac{-19 \pi}{6} + \frac{24 \pi}{6}| = \frac{5 \pi}{6}\).

Key Concepts

Radians

Radians are a way of measuring angles based on the radius of a circle. Rather than describing an angle in degrees, we use the length of an arc of the circle. The fundamental principle behind the radian is based on the idea that one full rotation around a circle is subtended by an angle of \(2\pi\) radians. This is equivalent to 360 degrees.

• 1 radian is approximately 57.296 degrees.
• A half circle or \(\pi\) radians is equal to 180 degrees.

Using radians is highly useful in mathematics and physics as they provide a natural way to relate angle measurements to real-world applications. When solving problems involving angles, converting between degrees and radians helps us apply formulas more effectively. It's often easier to manipulate trigonometric functions in radians due to their direct relationship with unit circles.

Quadrant

A quadrant refers to each of the four sections of the coordinate plane. These sections are divided by the x-axis and y-axis. Knowing which quadrant an angle falls into is crucial because it affects the sign and value of the trigonometric functions associated with that angle.

• First Quadrant: Where both x and y values are positive.
• Second Quadrant: x values are negative, y values are positive.
• Third Quadrant: Both x and y values are negative.
• Fourth Quadrant: x values are positive, y values are negative.

For the angle \(-\frac{7 \pi}{6}\), we first consider its negative measurement. Negative angles are measured clockwise from the positive x-axis. This places \(-\frac{7 \pi}{6}\) in the fourth quadrant. Understanding this placement helps in identifying the reference angle and understanding the behavior of sine, cosine, and tangent functions for angles in this quadrant.

Angle Measurement

Angle measurement is the process of determining the size of an angle in a specific unit, such as degrees or radians. Accurately measuring angles is critical for understanding geometry and trigonometry, where the relationship between angles and side lengths of triangles is essential.

• In degrees, a circle measures 360 degrees.
• In radians, a full circle measures \(2\pi\) radians.

For determining reference angles, understanding angle measurement allows you to take any angle and calculate the smallest angle it makes with the x-axis. In mathematical problems, this concept helps simplify complex angle calculations by bringing the problem into a more manageable form. Calculating reference angles typically involves converting the angle into a positive measure that can be evaluated with respect to the x-axis.