Problem 7 - Basic Technical Mathematics with Calculus

Question

Draw the given angles. $$50^{\circ},-120^{\circ},-30^{\circ}$$

Step-by-Step Solution

Answer

50° is counterclockwise; -120° and -30° are clockwise.

Step 1: Understanding the Positive Angle

To draw a positive angle like \(50^{\circ}\), start from the positive x-axis (the right side of the horizontal line) and rotate counterclockwise. The angle \(50^{\circ}\) means you rotate 50 degrees in the counterclockwise direction. Draw this angle in standard position starting from the positive x-axis.

Step 2: Understanding a Negative Angle

For negative angles, such as \(-120^{\circ}\), rotation is done clockwise. Begin at the positive x-axis and rotate clockwise to reach the desired angle. For \(-120^{\circ}\), imagine starting from zero and moving clockwise 120 degrees to reach the final position of the angle.

Step 3: Drawing the -30° Angle

To draw \(-30^{\circ}\), start again from the positive x-axis, and then rotate clockwise by 30 degrees. This slightly less than quarter circle brings the terminal side below the x-axis in the fourth quadrant.

Key Concepts

Positive Angles

A positive angle is formed by a counterclockwise rotation starting from the positive x-axis. These angles indicate a turn in the direction opposite to that of a clock's hands. For example, a positive angle of \(50^{\circ}\) means that you begin from the right side of the horizontal axis and rotate upwards along a circular arc. Positive angles range from \(0^{\circ}\) to \(360^{\circ}\) in a single full rotation.

To draw a \(50^{\circ}\) angle:

• Start on the positive x-axis.
• Measure 50 degrees counterclockwise from this starting point.
• The terminal side will be slightly above the 0-degree line in the first quadrant.

Visualizing positive angles as counterclockwise arcs makes drawing them easier and helps understand their position in a coordinate system.

Negative Angles

Negative angles involve rotation in a clockwise direction, starting from the positive x-axis. They typically indicate a rotation that goes in the same direction as a clock's hands. For example, when dealing with a negative angle like \(-120^{\circ}\), the movement is downward from the starting line.

Here's how to interpret a \(-120^{\circ}\) angle:

• Begin at the positive x-axis direction (the familiar starting point for angle measurement).
• Rotate clockwise for 120 degrees.
• The terminal side of this angle will rest within the third quadrant.

Negative angles are a bit counterintuitive initially, but once you practice, you'll find them easier to visualize.

Clockwise Rotation

Clockwise rotation implies turning a figure in the direction that the hands on a clock move. For drawing angles, it's particularly relevant when dealing with negative angles. When you have an angle labeled with a negative sign, like \(-30^{\circ}\), you'll make a clockwise movement from the positive x-axis.

To create a \(-30^{\circ}\) angle:

• Start on the positive x-axis.
• Rotate 30 degrees in a clockwise direction.
• This positions the terminal side in the fourth quadrant, just below the horizontal axis.

Remember that any negative angle is always coupled with a clockwise turn, which simplifies drawing and understanding.

Counterclockwise Rotation

Counterclockwise rotation points in a direction opposite to that of a clock's hands. It's the default orientation for positive angles. This means if an angle has no negative sign, it rotates counterclockwise from the positive x-axis.

Understanding a counterclockwise rotation:

• Begins at the positive x-axis.
• Moves upward through the quadrants based on the degree of the angle.
• For example, a \(50^{\circ}\) angle moves slightly into the first quadrant.

This kind of rotation helps in mapping out angles accurately and is at the heart of angle measurement in trigonometry and geometry.