Problem 61 - Precalculus

Question

State in which quadrant or on which axis each angle with the given measure in standard position would lie. $$-540^{\circ}$$

Step-by-Step Solution

Answer

The angle \(-540^{\circ}\) lies on the negative x-axis.

Step 1: Understand the Problem

We need to determine in which quadrant or on which axis the angle \(-540^{\circ}\) lies when in standard position. Angles in standard position start from the positive x-axis and move counter-clockwise.

Step 2: Find the Equivalent Positive Angle

Since \(-540^{\circ}\) is a negative angle, we can add \(360^{\circ}\) repeatedly to find an equivalent positive angle. Start by adding \(360^{\circ}\) once: \(-540^{\circ} + 360^{\circ} = -180^{\circ}\). Since this angle is still negative, add \(360^{\circ}\) again: \(-180^{\circ} + 360^{\circ} = 180^{\circ}\).

Step 3: Determine Angle Position

Now we determine the position of the angle \(180^{\circ}\) in standard position. An angle of \(180^{\circ}\) is on the negative x-axis.

Key Concepts

Standard Position

Angles are often measured in what is known as the "standard position". This is a way to visualize angles on the coordinate plane. An angle is in standard position when its vertex is at the origin and its initial side lies along the positive x-axis. The direction in which the angle is measured determines whether it is positive or negative. Standard position helps in providing a consistent way to describe where angles lie on the plane.
Positive angles are typically measured counter-clockwise from the initial side. If you imagine standing on the positive x-axis and turning in a circle, turning to your left is considered the positive direction. Understanding this initial setup is essential because it allows for clear communication and interpretation of angles.

Quadrants

The coordinate plane is divided into four sections called quadrants. They are numbered in a counter-clockwise direction:

• Quadrant I: where both x and y are positive.
• Quadrant II: where x is negative and y is positive.
• Quadrant III: where both x and y are negative.
• Quadrant IV: where x is positive and y is negative.

Knowing in which quadrant an angle terminates is helpful. It aids in calculating trigonometric values, determining directions, and solving geometry problems. For example, an angle between 0° and 90° would be in Quadrant I, while one between 90° and 180° would lie in Quadrant II. However, angles such as 180° or 360° lie on an axis rather than within a quadrant.

Negative Angles

Negative angles are simply angles measured in the clockwise direction from the positive x-axis. This may sound confusing, but it's just the opposite of the standard counter-clockwise direction. For example, an angle of \(-90^{\circ}\) signifies a turn to the right instead of left.
Negative angles can be converted to positive equivalents by adding \(360^{\circ}\) until achieving a positive measure. This is because the coordinate plane loops every \(360^{\circ}\). In our exercise, adding \(360^{\circ}\) once turned \(-540^{\circ}\) into \(-180^{\circ}\), and adding again resulted in \(180^{\circ}\), showing how you can loop around the plane to find equivalent angles.

Coordinate Axes

Angles can also lie along the axes themselves. When an angle's terminal side lies directly on an axis, it doesn't belong to any quadrant. Instead, it rests on one of these distinct positions:

• The positive x-axis: \(0^{\circ}\)
• The positive y-axis: \(90^{\circ}\)
• The negative x-axis: \(180^{\circ}\)
• The negative y-axis: \(270^{\circ}\)

In the problem, after converting \(-540^{\circ}\) to \(180^{\circ}\), we see that it lies on the negative x-axis. Recognizing angles on the axes is important because they represent key reference points in many mathematical and real-world contexts.