Problem 63 - Precalculus: A Concise Course

Question

Rewrite each angle in degree measure. (Do not use a calculator.) (a) \(\frac{5 \pi}{4}\) (b) \(-\frac{7 \pi}{3}\)

Step-by-Step Solution

Answer

Therefore, \(\frac{5 \pi}{4}\) radians is equivalent to 225 degrees and \(-\frac{7 \pi}{3}\) radians is equivalent to -420 degrees

Step 1: Convert \(\frac{5 \pi}{4}\) to degrees

To convert \(\frac{5 \pi}{4}\) radians to degrees, multiply \(\frac{5 \pi}{4}\) by the equivalent of \( \pi \) in degrees which is 180 degrees. Therefore you get \(\frac{5 \pi}{4} \times \frac{180}{\pi}\). Here \( \pi \) would cancel out and only the numerical conversions remains. This is equated to \(\frac{5 \times 180}{4} = 225\) degrees

Step 2: Convert \(-\frac{7 \pi}{3}\) to degrees

Converting \(-\frac{7 \pi}{3}\) to degrees, similar to the first step, multiply \(-\frac{7 \pi}{3}\) by the equivalent in degrees of \( \pi \) radians which is 180 degrees. Hence \(-\frac{7 \pi}{3} \times \frac{180}{\pi}\). Again \( \pi \) would cancel out and we have \(-\frac{7 \times 180}{3} = -420\) degrees

Key Concepts

Radians to Degrees

Converting angles from radians to degrees is a fundamental concept in trigonometry and precalculus. Radians and degrees are two different units for measuring angles, much like a pound and a kilogram for weight. Understanding how to convert between them is crucial because some mathematical applications prefer one over the other. To convert radians to degrees, you need to know that \( \pi \ \approx 180^\circ\). This means one radian is equivalent to approximately 57.2958 degrees.
Therefore, the conversion process is straightforward:

• Multiply the radian measure by \( \frac{180}{\pi}\).

For example, given \(\frac{5\pi}{4}\), multiply by \( \frac{180}{\pi}\), leading to \(225^\circ\). The \(pi\)'s cancel out, simplifying the process.
Alternatively, for \( -\frac{7\pi}{3}\), multiply by \( \frac{180}{\pi} \) to obtain \(-420^\circ\), recognizing that minus signs convey direction.

Trigonometry

Trigonometry is the branch of mathematics that deals with the study of triangles, specifically right triangles, and the relationships between their angles and sides. In this context, angles in trigonometry are often measured in radians, especially in more advanced mathematics.
Trigonometric functions such as sine, cosine, and tangent are essential for describing the properties of triangles. These functions also repeat every \(2\pi\) radians because the circle, central to trigonometry, contains \(2\pi\) radians. Thus, understanding angle conversion helps in solving trigonometric problems.

• Remember that trigonometric identities often require angles in radians or degrees depending on the problem context.
• Mastering conversions between radians and degrees fosters a deeper understanding of circular functions and periodic phenomena.

Using radians simplifies some formulas, particularly in calculus, but degrees are often more intuitive.

Precalculus

Precalculus serves as the bridge between algebra, trigonometry, and calculus. It provides the necessary foundation for understanding the behavior of functions that model real-world phenomena. Angle conversion, particularly from radians to degrees, is an essential skill in precalculus.
This conversion facilitates the transition from high school algebra and geometry to higher-level mathematics involving complex equations and functions.

• Precalculus provides a preview of concepts such as derivatives and integrals, which heavily rely on radians for angular measurements.
• Key topics include functions, complex numbers, conic sections, vectors, and of course, trigonometry.

Understanding how angles and their measures connect to these broader mathematical concepts is crucial as students progress to calculus. By mastering angle conversions, students enhance their problem-solving tools, making them better equipped for future mathematical challenges.