Question

Convert each angle in degrees to radians. Express your answer as a multiple of \(\pi .\) \(-225^{\circ}\)

Short answer

-225° = \(-\frac{5\pi}{4}\) radians

Step-by-step solution

Understand the Conversion Factor

To convert degrees to radians, use the conversion factor \ \( \frac{\pi}{180^{\circ}} \). This is derived from the fact that \(180^{\circ}\) is equivalent to \(\pi\) radians.

Set Up the Conversion

Multiply the angle in degrees by the conversion factor. For this problem: \ \(-225^{\circ} \times \frac{\pi}{180^{\circ}} \)

Simplify the Expression

First, calculate the multiplication of the constants: \ \(-225 \times \frac{\pi}{180} = -225 \times \frac{\pi}{180} = \frac{-225\pi}{180} \). Then, simplify the fraction: \ \frac{-225\pi}{180} = \frac{-5\pi}{4}\

Write the Final Answer

Therefore, \(-225^{\circ}\) in radians is expressed as \(\frac{-5\pi}{4} \).

Key concepts

angle measurement

Angles are measured in either degrees or radians. Degrees are more common in everyday use, such as measuring angles in a triangle or the hour divisions on a clock. One complete circle is 360 degrees. Radians, on the other hand, are often used in higher mathematics and physics. Here, one complete circle is defined as an angle of \(2\pi\) radians. To relate these two systems, it's useful to remember that 180 degrees is equivalent to \( \pi \) radians. This foundational relationship makes converting between degrees and radians straightforward.

radian

A radian is a way of measuring angles based on the radius of a circle. One radian is the angle created when the arc length along the circle's edge is equal to the radius of the circle. Essentially, if you took the radius of a circle and wrapped it along the circumference, the angle formed is one radian. This natural measurement is handy in many areas of math and science because, unlike degrees, it directly relates angles to lengths and circles. To visualize this, imagine placing the radius length along the outside edge of the circle; that's the arc length that corresponds to one radian.

conversion factor

To convert angles from degrees to radians, we use a simple multiplication factor: \(\frac{\pi}{180^{\circ}}\). This factor comes from the relationship that \( \pi \) radians are equivalent to 180 degrees. Whenever you have an angle in degrees, you multiply it by \( \frac{\pi}{180^{\circ}} \) to convert it to radians. For example, converting -225 degrees to radians involves: \(-225^{\circ} \times \frac{\pi}{180^{\circ}}\). This simple multiplication will give you the angle in radians. It’s vital to remember and use this conversion factor whenever you switch between degrees and radians.

simplifying fractions

After converting the angle from degrees to radians, you might end with a fraction that can be simplified. Simplifying fractions involves reducing the numerator and denominator to their smallest whole numbers. In the example of \(\frac{-225\pi}{180}\), you simplify it by finding the greatest common divisor (GCD) of 225 and 180, which is 45. Dividing both the numerator and the denominator by 45 yields \(\frac{-225\div45 \pi}{180\div45} = \frac{-5\pi}{4}\). The simplified fraction makes your final answer cleaner and easier to understand. Remember, simplifying is an important final step in ensuring your calculations are as neat and accurate as possible.