Problem 2 - Algebra and Trigonometry with Analytic Geometry

Question

If the given angle is in standard position, find two positive coterminal angles and two negative coterminal angles. (a) \(240^{\circ}\) (b) \(315^{\circ}\) (c) \(-150^{\circ}\)

Step-by-Step Solution

Answer

(a) Positive: 600°, 960°; Negative: -120°, -480°. (b) Positive: 675°, 1035°; Negative: -45°, -405°. (c) Positive: 210°, 570°; Negative: -510°, -870°.

Step 1: Understand Coterminal Angles

Coterminal angles are angles that share the same terminal side when drawn in standard position. We can find them by adding or subtracting multiples of 360 degrees from the given angle.

Step 2: Find Positive Coterminal Angles for 240°

Start with the given angle, 240°. Add multiples of 360° to find positive coterminal angles: First angle: 240° + 360° = 600° Second angle: 240° + 720° = 960°

Step 3: Find Negative Coterminal Angles for 240°

Start with the given angle, 240°. Subtract multiples of 360° to find negative coterminal angles: First angle: 240° - 360° = -120° Second angle: 240° - 720° = -480°

Step 4: Find Positive Coterminal Angles for 315°

Begin with the given angle, 315°. Add multiples of 360° to find positive coterminal angles: First angle: 315° + 360° = 675° Second angle: 315° + 720° = 1035°

Step 5: Find Negative Coterminal Angles for 315°

Use the given angle, 315°. Subtract multiples of 360° to find negative coterminal angles: First angle: 315° - 360° = -45° Second angle: 315° - 720° = -405°

Step 6: Find Positive Coterminal Angles for -150°

Begin with the given angle, -150°. Add multiples of 360° to convert to positive angles: First angle: -150° + 360° = 210° Second angle: -150° + 720° = 570°

Step 7: Find Negative Coterminal Angles for -150°

Start with the given angle, -150°. Subtract multiples of 360° to find more negative coterminal angles: First angle: -150° - 360° = -510° Second angle: -150° - 720° = -870°

Key Concepts

Understanding Angle Measurement

Angle measurement is essential in trigonometry and geometry. Angles are typically measured in degrees. Degrees are represented by the symbol \(^{\circ}\). A complete circle is divided into 360 degrees, much like dividing a pie into 360 equal slices. Each slice represents 1 degree.

• A Right Angle is \(90^{\circ}\)
• A Straight Angle is \(180^{\circ}\)
• A Complete Rotation or a circle is \(360^{\circ}\)

To measure an angle, you look at the number of degrees between its two arms. If you imagine a circle, each angle represents a segment of that circle. This helps in understanding relationships among different angles, such as coterminal angles.

Angles in Standard Position

To better understand or compare angles, it is often helpful to place them in what's called the "standard position." In standard position:

• The angle's vertex is at the origin of a coordinate plane.
• The initial side lies along the positive x-axis.

From this position, the angle's measure is determined by the rotation of its terminal side. By using the standard position, it becomes easier to determine properties like coterminal angles. This positional reference helps unify our understanding and comparison of angles, as they all originate from the same point.

Positive and Negative Angles

Angles can be positive or negative, depending on the direction we measure them. This concept is crucial for understanding how to find coterminal angles.

• **Positive Angles**: Formed by rotating the terminal side counterclockwise from the initial side.
• **Negative Angles**: Formed by rotating the terminal side clockwise from the initial side.

Imagine standing at the center of a clock: - Turning left (counterclockwise) adds to the angle measure. - Turning right (clockwise) subtracts from the angle measure. Through understanding the direction of angles, one can deduce coterminal angles by adding or subtracting full 360° rotations.

Degrees in a Circle

One important concept in angle measurement is that a full circle is 360 degrees. This is a foundational concept in geometry and trigonometry. The 360 degrees can be linked to historical waypoints where ancient counting systems related a whole circle to this number. It marks how we can represent full rotations and helps in calculating coterminal angles by letting us add or subtract 360 degrees as needed.

• To find positive coterminal angles, add 360 degrees.
• To find negative coterminal angles, subtract 360 degrees.

Understanding the relation of degrees to a circle also allows us to visualize rotations and comprehend the complete scope of an angle's reach.