Problem 15 - Algebra 2 and Trigonometry

Question

In \(8-17,\) name the quadrant in which an angle of each given measure lies. $$ -400^{\circ} $$

Step-by-Step Solution

Answer

The angle \(-400^{\circ}\) lies in the Fourth Quadrant.

Step 1: Prepare the Angle

First, we need to reduce the angle \(-400^{\circ}\) to an equivalent angle between \(0^{\circ}\) and \(360^{\circ}\). Since the angle is negative, we add \(360^{\circ}\) repeatedly until we reach a positive angle.

Step 2: Calculate the Equivalent Angle

Add \(360^{\circ}\) to \(-400^{\circ}\) to normalize the angle within one full revolution:\(-400^{\circ} + 360^{\circ} = -40^{\circ}\).Since \(-40^{\circ}\) is still negative, we add \(360^{\circ}\) once more:\(-40^{\circ} + 360^{\circ} = 320^{\circ}\).

Step 3: Determine the Quadrant

The equivalent angle is \(320^{\circ}\), which lies between \(270^{\circ}\) and \(360^{\circ}\). Therefore, this angle is in the Fourth Quadrant.

Key Concepts

Equivalent Angles

When working with angles, sometimes we end up with large or negative numbers. It's important to boil them down to a simpler form. An equivalent angle is one that has the same terminal side on a coordinate plane as the original angle, but fits within a standard range of values, typically from \(0^{\circ}\) to \(360^{\circ}\).
Understanding equivalent angles helps us determine the angle's position in a quadrant easily and simplifies many calculations.
For example, if you're given an angle of \(720^{\circ}\), you can repeatedly subtract \(360^{\circ}\) to reduce it, indicating that \(720^{\circ} = 0^{\circ}\). This means \(720^{\circ}\) is equivalent to \(0^{\circ}\).
Keep in mind that the number of times you revolve around the circle (once or multiple times) does not change the angle's quadrant position.

Negative Angles

Typically, angle measures are positive, but there are instances where you encounter negative angles. Negative angles rotate clockwise on a coordinate plane, opposite to the usual counterclockwise direction of positive angles. Understanding this concept is vital to decoding such problems.
To convert a negative angle into a positive equivalent angle between \(0^{\circ}\) and \(360^{\circ}\), you can keep adding \(360^{\circ}\).

• Example: Given \(-45^{\circ}\), add \(360^{\circ}\) to get \(315^{\circ}\).
• Thus, \(-45^{\circ}\) and \(315^{\circ}\) are equivalent, both locating the terminal side in the Fourth Quadrant.

Mixing negative values into angle calculations offers another layer of complexity but these can be simplified by flipping them into their positive counterparts.

Angle Reduction

Angle reduction is a technique used to simplify angles by bringing them within the primary range of \(0^{\circ}\) to \(360^{\circ}\). It helps you comfortably identify which quadrant an angle lies in without getting tangled in excessive calculations.
The process is simple:

• If the angle is negative, add multiples of \(360^{\circ}\) until it becomes positive.
• If the angle exceeds \(360^{\circ}\), subtract \(360^{\circ}\) until it falls within the desired range.

Using the original exercise's example, take an angle of \(-400^{\circ}\). First, add \(360^{\circ}\) to get \(-40^{\circ}\). Since \(-40^{\circ}\) is still negative, add another \(360^{\circ}\) to arrive at \(320^{\circ}\).
Now, with \(320^{\circ}\), you can easily identify that it falls within the Fourth Quadrant, simplifying the process of deciding its quadrant position.