Problem 16 - Algebra and Trigonometry with Analytic Geometry

Question

Find the exact degree measure of the angle. (a) \(-\frac{5 \pi}{2}\) (b) \(9 \pi\) (c) \(\frac{\pi}{16}\)

Step-by-Step Solution

Answer

(a) -450° (or 270°), (b) 1620° (or 180° normalized), (c) 11.25°.

Step 1: Understand the Relationship Between Radians and Degrees

To convert radians to degrees, we use the relationship that \[ 2\pi \, \text{radians} = 360^\circ \] which simplifies to \[ \pi \, \text{radians} = 180^\circ \] Thus, each radian can be converted to degrees by multiplying by \( \frac{180}{\pi} \).

Step 2: Convert Part (a) from Radians to Degrees

We start by converting \[ -\frac{5 \pi}{2} \] to degrees by multiplying by \( \frac{180}{\pi} \).\[ -\frac{5 \pi}{2} \times \frac{180}{\pi} = -\frac{5 \times 180}{2} = -450^\circ \]

Step 3: Convert Part (b) from Radians to Degrees

For the angle \( 9\pi \), multiply to convert to degrees:\[ 9\pi \times \frac{180}{\pi} = 9 \times 180 = 1620^\circ \]

Step 4: Convert Part (c) from Radians to Degrees

The angle \( \frac{\pi}{16} \) in degrees is given by multiplying:\[ \frac{\pi}{16} \times \frac{180}{\pi} = \frac{180}{16} = 11.25^\circ \]

Step 5: Expressing Large or Negative Angles Within 0 to 360 Degrees

In some contexts, angles are best expressed within a single rotation, between \(0^\circ\) and \(360^\circ\). To "normalize" \(-450^\circ\), add \(360^\circ\) repeatedly until within this range:\[ -450^\circ + 360^\circ = -90^\circ \] which is equivalent to \(270^\circ\) when converted to positive angles within one rotation.For \(1620^\circ\), subtract \(360^\circ\) until within one full rotation:\[ 1620^\circ - 4 \times 360^\circ = 180^\circ \].

Key Concepts

Radian to Degree Conversion

To understand how to convert from radians to degrees, it's essential to know the basic relationship:

• The full circle in radians is measured as \(2\pi\) radians, equating to \(360^\circ\) in degrees.
• From this relationship, we derive \(\pi\) radians as equal to \(180^\circ\).

This implies that converting a radian measure to degrees involves multiplying the radian value by \( \frac{180}{\pi} \). Every single radian is equivalent to approximately \(57.2958^\circ\).

For example, if you have a radian measure like \(-\frac{5\pi}{2}\), simply multiply it by \(\frac{180}{\pi}\) to get the degree value \(-450^\circ\). This comes in handy when you encounter angles in trigonometry problems, making calculations and comprehension more intuitive in the familiar degree format.

Angle Normalization

Angle normalization involves expressing an angle within a specific and often more useful range.

For most practical purposes, angles are normalized within \(0^\circ\) to \(360^\circ\). This keeps the angle within one full rotation around the circle. It can be very useful in applications involving circular motion or trigonometric calculations where multiple rotations are irrelevant.

For instance, if you start with an angle of \(-450^\circ\), you might realize this angle includes multiple full rotations. To normalize, add \(360^\circ\) until the result is non-negative yet less than \(360^\circ\). This process would give you \(270^\circ\), since \(-450^\circ + 360^\circ = -90^\circ\), and \(-90^\circ\) plus another \(360^\circ\) equals \(270^\circ\).

Similarly, for an angle of \(1620^\circ\), subtract \(360^\circ\) repeatedly until it falls within our desired range: doing this four times leads to \(180^\circ\).

Angle Measurement

The measurement of angles is a foundational skill in trigonometry and plays an important role in various fields, such as physics, engineering, and computer graphics.

Angles define direction and orientation. They can be measured in different units, with degrees and radians being the most common. Understanding these measurements helps in transforming angles for different applications, like rotating shapes or analyzing wave functions.

• **Degrees**: Simplest, most common in day-to-day use, representing 1/360 of a full circle.
• **Radians**: Tying angle measure to arc length, where the angle in radians is the length of the arc that subtends that angle at the center of a circle, with radius as one. Therefore the full circle \(2\pi\) radians equivalent to \(360^\circ\).

These measurements let us describe phenomena that repeat cyclically, like wheel rotations or wave cycles, making them a key part of interpreting the world around us.