Problem 8 - Algebra 2 and Trigonometry

Question

In \(8-17,\) for each angle with the given degree measure, find the measure of the reference angle. \(100^{\circ}\)

Step-by-Step Solution

Answer

The reference angle is 80°.

Step 1: Understanding the Concept of Reference Angle

The reference angle is the acute angle formed by the terminal side of the given angle and the horizontal axis. For angles greater than 90° and less than 180°, the reference angle is calculated by subtracting the angle from 180°.

Step 2: Subtract from 180 Degrees

To find the reference angle for a given angle of 100°, we subtract 100° from 180°, as it lies in the second quadrant.\[ \text{Reference Angle} = 180^{\circ} - 100^{\circ} = 80^{\circ}\]

Key Concepts

Terminal Side

The terminal side of an angle is a crucial concept in trigonometry. When we talk about angles in relation to the Cartesian plane, the terminal side is the one that rotates from the initial side to form the angle. The initial side is often along the positive x-axis. The direction and position of the terminal side help determine the angle's properties.

• In our example, the angle is measured from the positive x-axis.
• The terminal side of this angle will extend into the second quadrant when dealing with angles between 90° and 180°.

This perspective is essential in trigonometry as it helps in visualizing and understanding how the angle interacts with the coordinate plane. When you consider the position of the terminal side, it not only aids in defining the quadrant the angle lies in but also helps in finding reference angles.

Acute Angle

In trigonometry, an acute angle is always less than 90 degrees. Reference angles are always acute, rendering them easier to work with in calculations and problem-solving scenarios.

• The beauty of a reference angle lies in its simplicity; no matter the size of your given angle, the reference angle is always positive and acute.
• For instance, if the given angle is 100°, we first check which part of the plane it lies in.
• Since it is between 90° and 180°, it's located in the second quadrant, so we subtract it from 180°.

This subtraction turns it into an acute angle which is easier to understand and work with while solving trigonometric problems. The reference angle of 100° becomes 80°, which fits perfectly into our acute angle definition.

Second Quadrant

The second quadrant of the Cartesian plane encompasses angles that range from 90° to 180°.

• Angles in the second quadrant can be tricky because they are neither acute nor right by themselves.
• The second quadrant is significant because while the sine values remain positive, cosine and tangent values become negative.

This quadrant tendency plays a vital role in defining the properties and behavior of the given angle. Knowing that the angle of 100° lies in this quadrant helps us determine its reference angle by subtraction from 180°. This insight not only confirms the reference angle as 80°—an acute angle—but also tells us about the general sign and behavior of trigonometric functions associated with it in this specific quadrant.