Problem 65 - Algebra and Trigonometry Real Mathematics, Real People

Question

Rewrite each angle in radian measure as a multiple of \(\pi .\) (Do not use a calculator.) (a) \(18^{\circ}\) (b) \(-240^{\circ}\)

Step-by-Step Solution

Answer

The radian measures equivalent to \(18^{\circ}\) and \(-240^{\circ}\) are \(\pi /10\) and -\(\frac{4\pi}{3}\) respectively.

Step 1: Conversion formula

Identify the conversion formula from degree to radians which is equal to \(\pi /180^{\circ}\) .

Step 2: Conversion of \(18^{\circ}\)

Convert \(18^{\circ}\) to radian measure by multiplying it with the conversion formula : \(18^{\circ} * \(\pi /180^{\circ}\) = \(\pi /10\).

Step 3: Conversion of \(-240^{\circ}\)

Convert \(-240^{\circ}\) to radian measure by multiplying it with the conversion formula: \(-240^{\circ} * \(\pi /180^{\circ}\) = -\(\frac{4\pi}{3}\).

Key Concepts

Degree to Radian Formula

The Degree to Radian Formula is essential when converting angle measurements from degrees to radians. This formula helps us translate these two ways of measuring angles: degrees, a historical system, and radians, which relates more closely to the properties of circles. The formula is straightforward: to convert degrees to radians, multiply the degree measure by \( \pi/180^{\circ} \). This works because a full circle, or 360 degrees, is equivalent to \( 2\pi \) radians. Thus, the ratio \( \pi/180^{\circ} \) converts any angle from degrees to radians.

To apply it practically, let's take an example: say you need to convert \( 45^{\circ} \) into radians. You would calculate \( 45 \times \frac{\pi}{180} = \frac{\pi}{4} \) radians. This transformation is straightforward yet foundational for learning more advanced concepts.

Angle Measurement

Understanding how angle measurement works is critical in mathematics, especially in trigonometry and geometry. Angles can be measured in two primary ways: degrees and radians. Each method offers unique insights and applications.

Degrees are probably the most familiar to you, dividing a circle into 360 equal parts. This straightforward method helps when relating angles to turns or slices of a circle. On the other hand, radians offer a mathematical perspective focusing on the arc's length relative to the circle's radius. Since a full circle encompasses \( 2\pi \) radians, an angle's measure in radians represents the measure of an arc cut out by that angle.

Accurately measuring angles is crucial for solving geometric problems, working with periodic functions, and understanding rotational dynamics. Choosing between degrees and radians depends on the application: use degrees for practical tasks and radians for mathematical computations.

Radians vs Degrees

Radians and degrees are both units used to measure angles. While degrees are more intuitive and commonly used in everyday situations, radians are more mathematically significant, especially in advanced mathematics. Understanding the difference between these two is important for transitioning between basic and advanced mathematics.

Degrees constitute a smaller, easily quantifiable unit within a full circle, with 360 degrees making it simple for practical use like navigation and construction. Radians, however, are based on the mathematical constant \( \pi \). Because the circumference of a unit circle is \( 2\pi \), a radian is defined as the angle produced when this circumference is wrapped around the circle's radius.

Radians provide a natural, inherent measure for angles in mathematical analysis, especially in calculus where functions such as sine and cosine are naturally periodic with periods that are fractions of \( \pi \). When you learn to convert between these two units seamlessly, you can navigate both simple and complex mathematical concepts with ease.