Problem 42 - Precalculus Mathematics for Calculus

Question

Find an angle between \(0^{\circ}\) and \(360^{\circ}\) that is coterminal with the given angle. $$-100^{\circ}$$

Step-by-Step Solution

Answer

The angle 260° is coterminal with -100°.

Step 1: Understand Coterminal Angles

Coterminal angles are angles that share the same initial and terminal sides. To find a coterminal angle, we can add or subtract full rotations (360°) from the given angle.

Step 2: Add 360°

Since the given angle is negative, we add 360° to find a positive coterminal angle that falls within the range of 0° to 360°.\[-100^{\circ} + 360^{\circ} = 260^{\circ}\]

Step 3: Verify the Angle

Check that the angle 260° is within the specified range of 0° to 360°. Since 260° falls within this range, it is a valid coterminal angle.

Key Concepts

Angle Measurement

In geometry, angle measurement is typically expressed in degrees, which is a way to describe the size of an angle formed by two rays sharing a common endpoint.

• A full circle is divided into 360 equal parts, each part representing one degree (\(^\circ\)).
• Other units of angle measurement include radians, but degrees are often used in everyday language and calculations involving basic geometry.

Angles can be positive or negative depending on the direction of rotation:

• Positive angles result from counterclockwise rotation, which is the standard direction for measuring angles.
• Negative angles result from clockwise rotation, indicating a reverse direction.

When dealing with angle measurement, understanding how angles relate to the circle is key. Whether the angle is positive or negative only affects the direction, not the position of the terminal side if the rotation is complete. This concept leads us to exploring coterminal angles.

Degree Rotation

Degree rotation describes the movement of an angle around a circle. A full rotation is 360 degrees, meaning if an angle rotates 360° from its starting point, it returns to its original position. This is due to completing one full cycle around the circle.

• Each 360-degree addition or subtraction leads the angle's terminal side to the same final position.

Coterminal angles are defined through this principle of degree rotation:

• When two angles are coterminal, their measures differ by a multiple of 360°.
• Adding or subtracting 360° from an angle allows us to find other angles that are coterminal.

For example, if you have an angle of \(-100^{\circ}\), adding 360° could give you the positive equivalent, \(260^{\circ}\), within the range of \(0^{\circ}\) to \(360^{\circ}\). This use of complete rotations helps in determining equivalent angle measures, especially in the context of trigonometry and navigation.

Negative Angles

Negative angles are those measured in a clockwise direction, opposite to the standard counterclockwise direction for angles traditionally.

• They can often be converted to positive angles by adding full 360° rotations.

Converting negative angles is important when trying to find coterminal angles that lie within a specific range, typically \(0^{\circ}\) to \(360^{\circ}\). For example, starting with a \(-100^{\circ}\) angle and adding \(360^{\circ}\) results in \(260^{\circ}\).

• This resulting angle, \(260^{\circ}\), remains within the preferred positive angle range.

Understanding how negative angles relate to positive angles is crucial in various practical scenarios, such as synchronizing mechanical systems, interpreting navigational data, or solving trigonometric problems.Always remember that the negative sign indicates only directionate rotation—it doesn't inherently change the magnitude of the angle.