Problem 32 - Algebra 2

Question

In which quadrant, or on which axis, does the terminal side of each angle lie? \(-\frac{5 \pi}{4}\) radians

Step-by-Step Solution

Answer

The terminal side of the angle \(-\frac{5 \pi}{4}\) radians lies in the third quadrant.

Step 1: Convert the angle to degrees

To gain a clearer understanding of where the provided angle lies, it can be converted to degrees. Since one full rotation (2\(\pi\) radians) is equivalent to 360 degrees, the conversion can be done by multiplying the given radian by \(\frac{180}{\pi}\). So, \(-\frac{5 \pi}{4}\) radians is \(-\frac{5 \pi}{4} * \frac{180}{\pi} = -225\) degrees.

Step 2: Determine the quadrant

Now, we draw a coordinate system and start at the positive part of the x-axis. A negative angle value means we turn clockwise on the coordinate system. A full rotation in clockwise direction would be -360 degrees. Since we only have -225 degrees, we stop short of a full rotation. Therefore, we end up in the third quadrant.

Key Concepts

Radian to Degree Conversion

Radian measures are often converted into degrees to make it easier to visualize angles on the coordinate plane. This is especially helpful when you're trying to locate where an angle is situated. To convert radians to degrees, you can use the formula:

• Degrees = Radians × \( \frac{180}{\pi} \)

This formula works because a full circle in radians is \(2\pi\), which is equivalent to 360 degrees. So each radian equals \(\frac{180}{\pi}\) degrees. For example, to convert \(-\frac{5\pi}{4}\) radians to degrees, you simply multiply:\[-\frac{5\pi}{4} \times \frac{180}{\pi} = -225 \text{ degrees}\]This conversion helps us determine where this angle's terminal side lies on the coordinate plane.

Negative Angle Measurement

When dealing with angles, it's important to understand how negative angles are measured. Most commonly, angles are measured starting from the positive x-axis. Positive angles move counterclockwise, while negative angles move clockwise.
If we consider an angle of \(-225\) degrees, it means you start from the positive x-axis and rotate 225 degrees in a clockwise direction. This method of measuring allows us to locate angles easily even if they are negative.

• Positive angles: Counterclockwise
• Negative angles: Clockwise

By knowing how the negative angle measurement works, we can easily chart it on a coordinate plane and determine its position relative to the axes.

Trigonometric Quadrants

The coordinate plane is divided into four quadrants, each having its own characteristics when it comes to angles:

• Quadrant I: Both x and y coordinates are positive.
• Quadrant II: x is negative, y is positive.
• Quadrant III: Both x and y coordinates are negative.
• Quadrant IV: x is positive, but y is negative.

To determine where an angle lies, start from the positive x-axis. For negative angles, as in \(-225\) degrees, rotate clockwise from the x-axis.
In our example, \(-225\) degrees brings us into the Third Quadrant. This quadrant is characterized by having both x and y coordinates as negative, crucial for understanding the sign of trigonometric functions like sine and cosine in different quadrants.