Problem 16 - Precalculus: Functions and Graphs

Question

Exer. \(13-16:\) Find the exact degree measure of the angle. $$\text { (a) }-\frac{5 \pi}{2} \quad \text { (b) } 9 \pi \quad \text { (c) } \frac{\pi}{16}$$

Step-by-Step Solution

Answer

(a) -450°, (b) 1620°, (c) 11.25°.

Step 1: Convert Radians to Degrees for Part (a)

We know that to convert from radians to degrees, we multiply by \( \frac{180}{\pi} \). Here, the angle is \(-\frac{5\pi}{2}\). Therefore, the degree measure is:\[-\frac{5\pi}{2} \times \frac{180}{\pi} = -450^\circ.\]

Step 2: Convert Radians to Degrees for Part (b)

Use the same conversion factor, \( \frac{180}{\pi} \), to convert \(9\pi\) to degrees:\[9\pi \times \frac{180}{\pi} = 1620^\circ.\]

Step 3: Convert Radians to Degrees for Part (c)

Again, use the conversion factor \( \frac{180}{\pi} \). For \(\frac{\pi}{16}\):\[\frac{\pi}{16} \times \frac{180}{\pi} = \frac{180}{16} = 11.25^\circ.\]

Key Concepts

Angle Measurement

Angles are crucial in understanding geometry and trigonometry. They are measured in two primary units: degrees and radians. A complete circle is equal to 360 degrees or \(2\pi\) radians. Every angle can be converted between these two systems using the conversion formulas.

• Degrees: A degree is one 360th of a full circle. This system is very popular in daily life scenarios, like in directions and navigation.
• Radians: This is the standard unit of angular measure used in many areas of mathematics. With radians, we use the arc length relative to the radius of a circle.

Understanding the relationship between these two helps in solving a variety of mathematical problems efficiently.

Trigonometry

Trigonometry deals with the properties and relations of angles and sides in a triangle. It's a branch of mathematics that is vital in fields such as astronomy, physics, engineering, and more. Understanding trigonometry often involves using angle measurements heavily.

• Basic functions like sine, cosine, and tangent define the relationships between angles and sides in a right triangle.
• These functions often require inputs in radians for precise calculations, although they can work with degrees as well.
• Trigonometry utilizes angles to solve real-world problems like finding distances and heights.

For successful applications in trigonometry, it's essential to master angle conversion between radians and degrees.

Mathematical Conversion

Conversion between radians and degrees is a fundamental skill. The key to converting angles from radians to degrees lies in using the conversion factor \( \frac{180}{\pi} \). This factor arises from the relationship: one full circle equals \(360\degree\) in degrees and \(2\pi\) in radians.

• To convert radians to degrees, multiply the radian by \(\frac{180}{\pi}\).
• Conversely, to convert degrees to radians, multiply the degree by \(\frac{\pi}{180}\).
• Conversion is useful because different fields and scholars might prefer one unit over the other.

Practicing these conversions with examples helps solidify understanding and boosts confidence in handling various mathematical tasks.