Problem 19 - Algebra and Trigonometry

Question

In Exercises \(13-20,\) convert each angle in degrees to radians. Express your answer as a multiple of \(\pi\). $$ -225^{\circ} $$

Step-by-Step Solution

Answer

-225 degrees is the same as \(-5\pi/4\) radians when expressed as a multiple of \(\pi\).

Step 1: Understand the Conversion Rate

First, recall that \(180^{\circ}\) is equal to \(\pi\) radians. Therefore, to convert degrees to radians, a conversion factor can be used, which is \(\pi/180\)

Step 2: Apply the Conversion Rate

Now, apply that conversion rate to the given angle. To do this, you simply multiply -225 by \(\pi/180\) to change the units from degrees to radians.

Step 3: Simplify the Solution

Next, simplify the resulting expression. In this case, -225 times \(\pi/180\) simplifies to \(-5\pi/4\). Since the problem asks for the answer to be expressed as a multiple of \(\pi\), no further simplification is necessary.

Key Concepts

Degrees to Radians

Angles are commonly measured in degrees and radians. When you want to convert an angle from degrees to radians, you're essentially changing the way the angle is measured. Degrees are part of a system that divides a circle into 360 equal parts, while radians are an alternative that measure an angle as the length of the arc on a unit circle. The full circle in radians is represented by the value of \(2\pi\), which is equivalent to \(360^{\circ}\). Thus, \(180^{\circ}\) is equal to \(\pi\) radians, which is a crucial piece of information for conversion. In our specific problem, the task was to convert \(-225^{\circ}\) to radians. With the right approach, this conversion becomes straightforward and manageable.

Conversion Factor

To convert degrees to radians, we use a conversion factor. The conversion factor is derived from the relationship we mentioned earlier: \(180^{\circ} = \pi\) radians. Thus, each degree is equivalent to \(\pi/180\) radians. Here's how it works:

• Identify the angle in degrees that you need to convert. For instance, \(-225^{\circ}\).
• Multiply the degree measure by the conversion factor \(\pi/180\) to transform it into radians.

This procedure is universal for any angle in degrees that you need to convert to radians, making it a valuable tool in your mathematical toolbox.

Simplification of Expressions

Once the initial conversion is performed by using the conversion factor, the resulting expression is often not in its simplest form. Simplifying this expression makes it easier to understand and use in subsequent calculations. In this exercise, you multiply \(-225\) by \(\pi/180\), resulting in \(-225\pi/180\).Simplification involves:

• Finding the greatest common divisor (GCD) of the numbers \(-225\) and \(180\). This helps reduce fractions to their simplest form.
• In this case, \(-225\) and \(180\) have a GCD of \(45\). Thus, the expression simplifies to \(-5\pi/4\).

By expressing the answer as a multiple of \(\pi\), we align with the typical format for radian measures, keeping it neat and efficient.

Radians

Radians are the standard unit of angular measure used in many areas of mathematics. Unlike degrees, radians relate angles directly to the geometry of a circle. A radian can be visualized as the angle created when you stretch a radius-length arc along the circumference of a circle.Some key points about radians:

• A complete circle encompasses \(2\pi\) radians.
• Radians can express angles more naturally in some mathematical contexts, particularly when it comes to calculus, trigonometry, and physics.
• By understanding radians, you'll find it easier to handle equations involving trigonometric functions and to work in general with angles in a more versatile way.

The conversion from degrees to radians opens up a new perspective in mathematical problems and allows for a deeper understanding of circular motion and periodic functions.