Problem 15 - Basic Technical Mathematics with Calculus

Question

Determine one positive and one negative coterminal angle for each angle given. $$278.1^{\circ}$$

Step-by-Step Solution

Answer

Positive: \(638.1^{\circ}\); Negative: \(-81.9^{\circ}\).

Step 1: Understanding Coterminal Angles

Coterminal angles are angles that share the same initial and terminal sides. They can be found by adding or subtracting multiples of \(360^{\circ}\) from the given angle.

Step 2: Finding a Positive Coterminal Angle

To find a positive coterminal angle of \(278.1^{\circ}\), add \(360^{\circ}\) to the angle. Thus, calculate \(278.1^{\circ} + 360^{\circ} = 638.1^{\circ}\).

Step 3: Finding a Negative Coterminal Angle

To find a negative coterminal angle, subtract \(360^{\circ}\) from \(278.1^{\circ}\). Therefore, compute \(278.1^{\circ} - 360^{\circ} = -81.9^{\circ}\).

Key Concepts

Positive Angle

A positive angle is one that is measured counterclockwise from the initial side to the terminal side. When you're determining coterminal angles, you often aim to add positive angles to your original measurement.
For example, starting with an angle of \(278.1^{\circ}\), a well-known way to find a positive coterminal angle is by adding \(360^{\circ}\), since it's a full rotation around a circle. This is how you get to \(638.1^{\circ}\).
While this angle \(638.1^{\circ}\) is greater than a full circle, it ends up in the same position on the unit circle as \(278.1^{\circ}\).

• Adding \(360^{\circ}\) helps find a coterminal angle in the positive direction.
• Positive angles are widely used since they give a sense of progress around a circle.

Negative Angle

A negative angle is simply the reverse session from the initial side to the terminal side, meaning it moves clockwise. This contrasts with how positive angles are measured. Finding a negative coterminal angle is as easy as subtracting \(360^{\circ}\) from the original angle.
For instance, if you start with \(278.1^{\circ}\) and subtract \(360^{\circ}\), you enter the negative realm with \(-81.9^{\circ}\).

• Negative angles are useful for calculations where the direction matters.
• They provide insight into how angles can be symmetric around a starting point.

With negative angles, you still end up at the exact same location on the unit circle but with a clockwise approach.

Angle Subtraction

Angle subtraction is an operation where you essentially take away a degree measure from an angle to find new representations, like in cases of coterminal angles.
In the original task, you took \(278.1^{\circ}\) and subtracted \(360^{\circ}\) to result in \(-81.9^{\circ}\).
This subtraction moves the angle into the negative quadrant or less than a full circle in a counterclockwise direction.

• Subtraction can help quickly determine negative coterminal positions.
• It assists in analyzing angles that are less than one full rotation.

Ensuring you understand angle subtraction is key to working seamlessly with angles in geometry and trigonometry.

Angle Addition

Angle addition, as the name suggests, involves adding angle measures together, a crucial skill for calculating positive coterminal angles.
By adding \(360^{\circ}\) to an existing angle, such as \(278.1^{\circ}\), you can find the coterminal angle that is also positive: \(638.1^{\circ}\). This approach continues the angle around a circle for one full rotation or turn.

• Using addition is a straightforward method to reposition angles in a more positive or forward rotation.
• Addition allows angles to complete more than one full rotation while still landing on the same terminal side.

Mastery of angle addition allows you to flexibly navigate through different circle segments and cycles, a necessary skill in various fields of math and physics.