Problem 16 - Algebra 2 and Trigonometry

Question

In \(8-17,\) for each angle with the given degree measure, find the measure of the reference angle. \(-130^{\circ}\)

Step-by-Step Solution

Answer

The reference angle is \(50^\circ\).

Step 1: Understanding Reference Angles

A reference angle is the smallest angle that the terminal side of an angle makes with the horizontal axis, and it is always a positive acute angle (less than \(90^\circ\)). For angles in different quadrants, reference angles are calculated differently, particularly if the angle is negative.

Step 2: Identify the Quadrant for the Angle

Since the given angle is \(-130^\circ\), we note that it is a negative angle, meaning it is measured clockwise. When we go \(-130^\circ\) from the positive x-axis, we find that this angle lies in the third quadrant.

Step 3: Calculate the Reference Angle for Third Quadrant

In the third quadrant, the reference angle is found by taking the negative angle and adding \(180^\circ\) to it. This is because angles in the third quadrant have their reference angle measured from the 180-degree line.\[\alpha_r = \text{absolute value of} (-130^\circ + 180^\circ)\]

Step 4: Compute the Calculation

Performing the calculation: \(-130^\circ + 180^\circ = 50^\circ\). The reference angle is \(50^\circ\), as it is already less than \(90^\circ\).

Key Concepts

Understanding Acute Angles

An acute angle is a special type of angle that is always less than \(90^{\circ}\). When we talk about the reference angle, we refer to the smallest angle made with the horizontal axis, which is always an acute angle.
An acute angle is typically well-known and very easy to identify.

• It can measure anywhere between \(0^{\circ}\) and just under \(90^{\circ}\).
• All reference angles, no matter the quadrant, are brought down to an acute angle.
• This helps keep calculations consistent and straightforward.

For instance, if you have a reference angle and see that it is \(50^{\circ}\), you immediately know it's acute because it's within the limits of \(0^{\circ}\) to \(90^{\circ}\). Keeping sight of the acute angle ensures that you stay within this valuable range for reference angles.

Navigating the Third Quadrant

Angles are categorized into four quadrants, and an angle's position within one of these quadrants dictates how to calculate its reference angle. The third quadrant is an interesting place for an angle to land.
Here's what happens when an angle lies there:

• The third quadrant is located between \(180^{\circ}\) and \(270^{\circ}\).
• When an angle lies in this quadrant, its reference is computed relative to \(180^{\circ}\).
• This means for any angle in the third quadrant, you find its reference angle by subtracting its measure from \(180^{\circ}\).

In the given example of a \(-130^{\circ}\), since it extends into the third quadrant, we use the method for this quadrant to make sense of the reference angle.

Understanding Negative Angles

Negative angles might initially seem a bit complex, but they follow the same rules as positive angles. Simply put, a negative angle is one that's measured clockwise from the positive x-axis. It’s the opposite of the more familiar counterclockwise measurement.
Here's a quick summary of dealing with them:

• Consider them as the mirror image of positive angles but in the opposite direction.
• Locate them accurately by subtracting their degree from \(360^{\circ}\) if necessary.
• For reference angle issues, convert them into a manageable counterpart, like we did with \(-130^{\circ}\).

Once correctly interpreted, negative angles seamlessly integrate into all angle tasks and shouldn't provide much trouble once you manage the initial conversion.

The Importance of Degree Measure

Understanding angles is crucial, and the first step is often understanding their measure. Degree measurement is how we commonly express angles on a circle, with each circle comprising \(360^{\circ}\).
Some essential elements of degree measure include:

• A single complete circle is equal to \(360^{\circ}\).
• We measure half a circle at \(180^{\circ}\), a useful point for third-quadrant reference angles.
• Degree measure lets us easily reference angles in terms the majority of us are familiar with from early math learning stages.

Every angle can be easily communicated in degree measure, providing a constant and reliable way to exchange or solve for angle relationships, as seen with finding reference angles.