Problem 35 - Precalculus Mathematics for Calculus

Question

The measures of two angles in standard position are given. Determine whether the angles are coterminal. $$\frac{5 \pi}{6}, \frac{17 \pi}{6}$$

Step-by-Step Solution

Answer

The angles \(\frac{5\pi}{6}\) and \(\frac{17\pi}{6}\) are coterminal because they differ by \(2\pi\).

Step 1: Understand Coterminal Angles

Two angles are coterminal if they share the same terminal side. This means the difference between the two angles should be a multiple of a full rotation, which is \(2\pi\) radians for angles given in radians.

Step 2: Calculate the Difference

Find the difference between the two angles: \( \frac{17\pi}{6} - \frac{5\pi}{6} \). First, subtract the fractions: \( \frac{17\pi}{6} - \frac{5\pi}{6} = \frac{17\pi - 5\pi}{6} = \frac{12\pi}{6} \).

Step 3: Determine Multiplicity of Full Rotation

Check if the result \( \frac{12\pi}{6} \) simplifies to a multiple of \(2\pi\). Simplifying gives \(2\pi\), which is exactly one full rotation.

Step 4: Conclusion

Since the difference is a full rotation (\(2\pi\)), the angles \(\frac{5\pi}{6}\) and \(\frac{17\pi}{6}\) are coterminal.

Key Concepts

Angle Measurements

In the world of geometry and trigonometry, angle measurements are crucial concepts. Angles can be measured in different units, and the two most common ones are degrees and radians.

• Degrees: A complete circle is 360 degrees. This system is often used in everyday life, like in navigation and construction.
• Radians: A complete circle is measured as \(2\pi\) radians. This measurement is frequently used in advanced mathematics because of its natural appearance in many mathematical formulas.

The angle measurement system you use can affect how you solve problems, especially when dealing with trigonometric functions or rotations. Understanding the behavior of angles in different measurement systems helps ensure precision and accuracy in calculations.

Radian Measure

Radian measure offers a different perspective compared to degrees. It is based on the radius of a circle. Simply put, one radian equals the angle made when the arc length is equal to the radius of the circle. If you imagine the circumference of a circle being a line that is rolled around the circle itself, that line is \(2\pi\) radians long.

• Relationship: One complete revolution of a circle is \(2\pi\) radians, equivalent to 360 degrees.
• Calculation: When dealing with radians, angles are typically expressed as multiples or divisions of \(\pi\), such as \(\frac{\pi}{2}\) or \(\frac{3\pi}{4}\).

This system is particularly advantageous in calculus given its natural ties to the unit circle and trigonometric identities. It simplifies certain calculations that would be more complex in degrees.

Full Rotation

A full rotation refers to turning a complete circle, returning to the starting position. This is significant when assessing coterminal angles.

• In degrees: A full rotation is 360 degrees.
• In radians: A full rotation is \(2\pi\) radians.

Coterminal angles are angles that terminate at the same point on a circle's circumference after a series of full rotations. By understanding full rotations, you can easily determine if two angles are coterminal. If the difference between two angles is a multiple of a full rotation (\(2\pi\) in radians), they will share the same terminal side. This principle helps in identifying angle properties, particularly in applications like trigonometry and circular motion.