Question

Convert each angle to degree measure. $$ \frac{9 \pi}{4} $$

Short answer

The angle in degree measure is \(405 ^\circ\).

Step-by-step solution

Write down the given angle in radians

We have the angle in radians as \(\frac{9\pi}{4}\).

Set up a proportion using the fact that π radians = 180 degrees

We'll set up the proportion: \[\frac{\frac{9\pi}{4}}{x} = \frac{\pi}{180}\]

Solve for x

To solve for x (the angle in degrees), we'll cross-multiply and divide: \(x = \frac{\frac{9\pi}{4}}{\pi} \times 180\)

Simplify the expression

First, the \(\pi\) terms cancel out: \(x = \frac{9}{4} \times 180\) Now, multiply the remaining fractions to find the angle in degrees: \(x = \frac{9 \times 180}{4}\) \(x = \frac{1620}{4}\)

Calculate x

Divide 1620 by 4 to find the angle in degrees: \(x = 405\) So, the angle in degree measure is \(\boxed{405 ^\circ}\).

Key concepts

Radian Measure

Understanding radians is crucial for angle calculations in various fields such as mathematics, physics, and engineering. Essentially, a radian is a unit of angular measure used in the International System of Units (SI). One full circle, which is 360 degrees in terms of degree measure, is equivalent to just over 6.28 radians, or more specifically, approximately \(2\pi\) radians. This is because the circumference of a circle is \(2\pi r\), where \(r\) is the radius.

Radian measure can be seen as the length of the arc on a unit circle, which is a circle with a radius of 1. One radian corresponds to an arc length exactly equal to the radius length. For instance, if you have an angle that subtends an arc of length \(r\) on a circle of radius \(r\), this angle is 1 radian. Hence, the relationship between radians and the parameters of a circle is quite direct and natural.

Degree Measure

Degree measure is another common way to express the size of an angle, widely used in daily life and various fields. A degree, symbolized by a small circle \( ^\circ \), represents \(1/360\) of a full rotation. Imagine slicing a pie into 360 equal slices – one slice represents one degree. The concept originates from the ancient Babylonians, who used a base-60 numbering system, hence the circle's division into 360 degrees.

Most people are familiar with degrees as they are used in weather forecasting, geography, and even in the setting of oven temperatures. Since degrees are a smaller subdivision of a circle compared to radians, they are often easier to visualize for small angles. However, when it comes to mathematical calculations, especially those involving trigonometric functions, radians are preferred due to their natural relation to the properties of a circle.

Angle Conversion

Converting between radians and degrees is a matter of applying the right conversion factor, helping to move between different fields and applications that may favor one unit over the other. The key relationship to keep in mind is that \(180^\circ\) is equivalent to \(\pi\) radians, stemming from the fact that a semi-circle creates an angle of \(180^\circ\) and its arc length is \(\pi\) times the radius.

When converting an angle from radians to degrees, you multiply the radian measure by \(\frac{180^\circ}{\pi}\). Conversely, to convert from degrees to radians, multiply the degree measure by \(\frac{\pi}{180^\circ}\). This straightforward calculation ensures that angles can be accurately converted regardless of the system of measurement being used.

Proportion Method

The proportion method is an effective strategy for angle conversion, as it leverages the fundamental relationship between radians and degrees. By setting up a proportion, one can easily calculate the unknown measure. Essentially, you create two ratios that capture this relationship and set them equal to each other. For example, if you have an angle \(a\) in radians, you can set up the proportion \(\frac{a}{x^\circ} = \frac{\pi}{180^\circ}\), where \(x^\circ\) is the unknown angle in degrees you wish to find.

Using cross-multiplication, you can then solve for \(x\). This method is not only helpful for angle conversion but is a solid example of how proportions are used to solve for unknown quantities in a wide range of mathematical problems. It emphasizes the beauty and utility of proportional reasoning in mathematics.