Problem 59 - Precalculus: A Concise Course

Question

Rewrite each angle in radian measure as a multiple of \(\pi\). (Do not use a calculator.) (a) \(-20^{\circ}\) (b) \(-60^{\circ}\)

Step-by-Step Solution

Answer

The angle \(-20^{\circ}\) is equivalent to \(-\frac{\pi}{9}\) radians. The angle \(-60^{\circ}\) is equivalent to \(-\frac{\pi}{3}\) radians.

Step 1: Convert \(-20^{\circ}\) to radians

To convert degrees to radians, use the fact that 180 degrees equals \(\pi\) radians. Therefore, to convert \(-20^{\circ}\) to radians, divide \(-20\) by 180 and multiply the result by \(\pi\). That simplifies to \(-\frac{20\pi}{180}\) which will further simplify to \(-\frac{\pi}{9}\).

Step 2: Convert \(-60^{\circ}\) to radians

Applying the same conversion factor as in Step 1, \(-60^{\circ}\) becomes \(-\frac{60\pi}{180}\). Upon simplifying, this results in \(-\frac{\pi}{3}\).

Key Concepts

Angle Conversion

Understanding angle conversion is essential in trigonometry and calculus, especially when switching between the degree and radian systems. Angles can be measured in various units, but the most common are degrees and radians. Each unit has its own advantages, and knowing how to convert between them allows for better flexibility in mathematical computations. Degrees, most commonly used in everyday situations, denote an angle based on dividing a circle into 360 parts. Radians, on the other hand, originate from the radius of a circle. To convert angles, it’s crucial to have a firm grasp of the relationship between these two measures. The key conversion factor is that 180 degrees is equivalent to \(pi\) radians. This translates to the formula:

• Degrees to radians: multiply by \( \frac{\pi}{180} \)
• Radians to degrees: multiply by \( \frac{180}{\pi} \)

Having this conversion factor lets us seamlessly move from one unit of measurement to the other. Being proficient in angle conversion allows students to tackle various mathematical problems effectively.

Radians

Radians are a revolutionary way to think about angles and can be more intuitive than degrees in certain mathematical contexts. They describe an angle based on the radius of a circle and the arc length it sub tends. Essentially, a radian measures angles through the direct relationship of the arc's length to the radius. One full circle in radians is exactly equal to \( 2\pi \) radians, corresponding to \( 360^\circ \). A semicircle, therefore, measures \( \pi \) radians. It simplifies many mathematical equations as it directly links algebraic operations with trigonometric concepts.The advantage of radians becomes apparent in calculus and higher mathematics. For example, differentiation and integration of trigonometric functions are often simpler when angles are in radians. This is because the radian measure reflects natural periodicity and symmetries of the circle.Understanding radians also aids in visualizing angles since they offer a geometric interpretation, making them a crucial component of trigonometry.

Degrees to Radians Conversion

Converting degrees to radians, as seen in our exercise, is a straightforward process once you understand the relationship between the units. Each degree can be represented as a fraction of \( \pi \) using the formula:\[\text{Radians} = \text{Degrees} \times \frac{\pi}{180}\]By applying this formula, let's look at the conversion of \( -20^\circ \) and \( -60^\circ \) to radians. For \( -20^\circ \), you multiply -20 by \( \frac{\pi}{180} \) which simplifies to \( -\frac{\pi}{9} \). Similarly, \( -60^\circ \) converts to \( -\frac{\pi}{3} \) by multiplying -60 by \( \frac{\pi}{180} \) and simplifying.This conversion is not only essential for solving problems that require switching between angle measurements but also ensures that mathematical solutions incorporate the direct geometric relationships inherent to radians. Practice using this conversion in various exercises to solidify your understanding and capability.