Problem 53 - Algebra 2

Question

Find one angle with positive measure and one angle with negative measure coterminal with each angle. \(\frac{17 \pi}{4}\)

Step-by-Step Solution

Answer

Positive coterminal angle: 1125°, Negative coterminal angle: -315°.

Step 1: Convert the Angle to Degrees

First, we need to convert the given angle from radians to degrees for easier handling. Use the conversion formula: \( \text{Degrees} = \frac{\pi}{180^\circ} \). Thus, \( \frac{17\pi}{4} \) in degrees is \( \frac{17\pi}{4} \times \frac{180^\circ}{\pi} = 765^\circ \).

Step 2: Find a Positive Coterminal Angle

Coterminal angles differ by full rotations (\(360^\circ\)). To find a positive coterminal angle, add \(360^\circ\) to \(765^\circ\): \(765^\circ + 360^\circ = 1125^\circ\).

Step 3: Find a Negative Coterminal Angle

For a negative coterminal angle, subtract \(360^\circ\) from \(765^\circ\): \(765^\circ - 360^\circ = 405^\circ\). Since 405° is still positive, subtract another \(360^\circ\) to get \(405^\circ - 360^\circ = 45^\circ\). For a truly negative coterminal angle, further subtract \(360^\circ\): \(45^\circ - 360^\circ = -315^\circ\).

Step 4: Conclusion on Coterminal Angles

Therefore, the angle \(1125^\circ\) (in degrees) is a positive coterminal angle, and \(-315^\circ\) is a negative coterminal angle to the original angle of \(\frac{17\pi}{4}\) radians.

Key Concepts

Radians to Degrees Conversion

Converting radians to degrees is an essential skill in angle measurement. In math and physics, radians are often used because they naturally relate to the circle's geometry. However, degrees are more intuitive for many people, which is why conversion is necessary. To convert radians to degrees, use the formula:

• Degrees = Radians × (180°/π)

This formula works because a complete circle is 360 degrees or 2π radians. So, each radian equals 180°/π. For example, to convert the angle \(\frac{17\pi}{4}\) radians into degrees:

• \(\frac{17\pi}{4}\) × \(\frac{180°}{\pi}\) = 765°

Knowing this conversion helps in understanding and performing more complex calculations like finding coterminal angles.

Angle Measurement

Angle measurement can be understood in two primary ways: degrees and radians. Each system offers different benefits. Degrees divide the circle into 360 equal parts. It's what's typically used in practical, day-to-day applications, like navigation. Radians, on the other hand, are based on the radius of the circle. They are the standard unit of angular measurement in calculus and many fields of advanced mathematics because they simplify mathematical formulas, especially in trigonometry and calculus.This dual system can seem complex at first, but understanding radians and degrees is crucial for transitioning between different areas of mathematics, as both are used frequently depending on the context. Whether converting between them or using them to solve problems, keep in mind how interrelated they are: 360° equals \(2\pi\) radians.

Positive and Negative Angles

Angles can be both positive and negative, which refers to the direction in which the angle is measured.

• Positive angles are measured counterclockwise from the positive x-axis.
• Negative angles are measured clockwise from the positive x-axis.

In a circle, each full rotation adds or subtracts 360° or \(2\pi\) radians, which means angles can be adjusted by these amounts to find coterminal angles. For instance, if you have the angle 765°, you can find a positive coterminal angle by adding 360° to get 1125°. For a negative coterminal angle, you subtract 360° multiple times, such as 765° - 360° - 360° = 45°, and finally 45° - 360° = -315°.Finding coterminal angles helps to determine equivalent angles that are more convenient for calculation or interpretation. This flexibility is especially useful in trigonometry and when dealing with periodic functions.