Problem 50 - A Graphical Approach to Precalculus with Limits

Question

Find the angle of least positive measure that is co terminal with the given angle. $$-98^{\circ}$$

Step-by-Step Solution

Answer

The angle of least positive measure coterminal with \(-98^{\circ}\) is \(262^{\circ}\).

Step 1: Understanding Coterminal Angles

Coterminal angles share the same terminal side, which means they differ by complete rotations of 360 degrees. To find a positive angle that is coterminal with \(-98^{\circ}\), we need to add 360 degrees until we reach a positive angle.

Step 2: Add 360 Degrees

Start by adding 360 degrees to the given angle: \(-98^{\circ} + 360^{\circ} = 262^{\circ}\). The resulting angle is positive.

Step 3: Verify the Angle

Ensure that the calculated positive angle \(262^{\circ}\) is a valid coterminal angle by considering if it is in the range from \(0^{\circ}\) to \(360^{\circ}\). Since \(262^{\circ}\) is within this range, it is the angle of least positive measure coterminal with \(-98^{\circ}\).

Key Concepts

Angle Measure

An angle is formed between two rays that share a common endpoint. The measure of an angle is a way to describe how "open" an angle is, and it is typically quantified in degrees. The complete circle is divided into 360 degrees, so when talking about angles, the measure indicates how much rotation is needed to get from one ray of the angle to the other ray.
Understanding angle measurement is key when working with angles in any mathematical context.

• Angles can range from 0° to 360° for a full rotation.
• A negative angle measure indicates a clockwise rotation, whereas a positive angle denotes counterclockwise rotation.
• When calculating an angle, ensure that you are always measuring in the right direction and that the total degree reflects the correct rotation.

In exercises involving coterminal angles, understanding how to measure and describe these angles correctly is crucial, as these involve identifying angles with shared terminal sides.

Positive Angle

A positive angle is one measured in a counterclockwise direction from the starting ray, known as the initial side, to the ending ray, called the terminal side.
It's often easier and more intuitive to work with positive angles, especially when communicating directions or rotations.

• Positive angles range from 0° to 360°.
• Such angles are frequently used in practical problems, including finding coterminal angles.
• A simple way to convert negative angles to positive angles is by adding 360° until the angle becomes positive.

In the given task, the negative angle \(-98^{\circ}\) was transformed into the positive angle \(262^{\circ}\) by adding \(360^{\circ}\). This effectively shifted the measurement to a counterclockwise direction while finding the smallest possible positive angle coterminal with the original angle.

Degree Calculation

When dealing with angles, particularly when trying to find coterminal angles, degree calculation becomes crucial. This involves appropriately adding or subtracting full rotations of \(360^{\circ}\) from your given angle.

• Adding \(360^{\circ}\) to a negative angle rotates the angle counterclockwise to yield a positive equivalent.
• To ensure the smallest positive coterminal angle, repeatedly add \(360^{\circ}\) until the angle is positive and falls within the desired range of \(0^{\circ}\) to \(360^{\circ}\).
• Conversely, if you have a large angle and need a smaller coterminal angle, you can subtract \(360^{\circ}\).

In solving the original exercise, the degree calculation was done by adding \(360^{\circ}\) to \(-98^{\circ}\), resulting in \(262^{\circ}\), a positive coterminal angle within the standard 0° to 360° range.