Problem 18 - Precalculus

Question

For the following exercises, draw an angle in standard position with the given measure. $$ -315^{\circ} $$

Step-by-Step Solution

Answer

Convert -315° to 45° and draw counterclockwise from the positive x-axis.

Step 1: Understand Angle in Standard Position

An angle in standard position is one whose vertex is at the origin (0,0) on the coordinate plane, and the initial side lies along the positive x-axis. Counterclockwise rotation defines positive angles, while clockwise rotation defines negative angles.

Step 2: Convert Negative Angle to Positive Equivalent

Since negative angles indicate clockwise rotation, we add multiples of 360° to find a standard position. For \(-315^{\circ}\), adding 360° gives us: \(-315^{\circ} + 360^{\circ} = 45^{\circ}\).

Step 3: Interpret the Result

The positive equivalent \(45^{\circ}\) tells us that the angle can be drawn by rotating counterclockwise 45 degrees from the positive x-axis. This results in an angle in the first quadrant.

Step 4: Draw the Angle

On graph paper or the coordinate plane, start at the positive x-axis. Rotate 45° counterclockwise to draw the angle. The terminal side will be in the first quadrant, creating a 45-degree angle with the x-axis.

Key Concepts

Negative Angles

When dealing with angles, you might come across negative values. What exactly do these negative angles represent? In essence, a negative angle signifies a direction of rotation. While most are familiar with counterclockwise rotation being the default (positive), negative angles imply the opposite: clockwise rotation.

• This means that if we have a negative angle, such as \(-315^{\circ}\), we start at the positive x-axis and rotate clockwise around the origin.
• Negative angles are essential in helping describe positions and movements that veer in the clockwise direction.

Knowing how to interpret negative angles is crucial in fields like physics and engineering, where understanding direction and rotation is pivotal.

Clockwise Rotation

Clockwise rotation is often how people picture the hands of a clock moving. When working with angles, it's valuable to apply this metaphor. In the context of mathematics:

• Clockwise rotation happens when you turn from the starting line (positive x-axis) in the direction of the clock's hands.
• This type of rotation is typically used for negative angles. For example, an angle of \(-315^{\circ}\) takes you 315 degrees in a clockwise fashion from the starting position.

Being able to visualize the direction of rotation helps in accurate angle drawing and in understanding the orientation of an object or line in relation to an angle.

Conversion to Positive Angles

Converting negative angles to positive equivalents can simplify understanding and drawing. The process involves a straightforward mathematical adjustment:

• Start with the negative angle you have, like \(-315^{\circ}\).
• Add 360° because a full circle (or rotation) is 360 degrees.
• The equation \(-315^{\circ} + 360^{\circ} = 45^{\circ}\) gives you the positive angle.

This conversion step reveals that \(\-315^{\circ}\) is equivalent to rotating 45 degrees counterclockwise from the positive x-axis. This process is key in finding how angles relate to each other through full rotations and lets you work with positive angles more consistently across problems and applications.