Problem 116 - Calculus Volume 1

Question

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

Step-by-Step Solution

Answer

The angle \(-225^{\circ}\) in radians is \(-\frac{5}{4}\pi\).

Step 1: Understand the Conversion Formula

To convert an angle from degrees to radians, use the conversion formula:\[\text{Radians} = \text{Degrees} \times \frac{\pi}{180}\]This formula comes from the fact that \(180^{\circ}\) is equivalent to \(\pi\) radians.

Step 2: Substitute the Given Degree Measure

Substitute the degree measure given in the problem, which is \(-225^{\circ}\), into the conversion formula:\[\text{Radians} = -225 \times \frac{\pi}{180}\]

Step 3: Simplify the Expression

Calculate \(-225 \times \frac{\pi}{180}\) by dividing \(-225\) by \(180\), which gives you:\[-225 \times \frac{1}{180} = \frac{-225}{180}\]Simplifying this fraction, by dividing both the numerator and the denominator by their greatest common divisor, which is \(45\), results in:\[\frac{-225}{180} = \frac{-5}{4}\]Thus, the expression becomes \(-\frac{5}{4}\pi\).

Step 4: Final Expression in Radians

The angle \(-225^{\circ}\) expressed as a multiple of \(\pi\) is:\[-\frac{5}{4}\pi\]Therefore, the final result is \(-\frac{5}{4}\pi\) radians.

Key Concepts

Degrees to Radians

When dealing with angles, you may encounter two primary measurement units: degrees and radians. While degrees are common in everyday use, radians often appear in mathematical contexts, particularly in calculus and trigonometry. Understanding both is essential for converting angles and solving various problems.
In simple terms, degrees divide a circle into 360 equal parts. On the other hand, radians use the circle's radius to measure angles. One full circle is equal to approximately 6.283 radians (or exactly \(2\pi\) radians).
Converting from degrees to radians allows us to take advantage of mathematical properties and formulas that use radians. This conversion becomes especially useful in trigonometric calculations, as angles in radians simplify the computation of trigonometric functions.

Conversion Formula

The conversion from degrees to radians is straightforward thanks to a specific formula. This formula stems from the relationship between the number of degrees and the number of radians in a full circle.
The formula is:

• Degrees to Radians: \[ \text{Radians} = \text{Degrees} \times \frac{\pi}{180}\]

This formula derives from the fact that 180 degrees equal \(\pi\) radians.
By multiplying the degree measurement by \(\frac{\pi}{180}\), you convert it into radians. This ensures the angle is expressed in terms of \(\pi\), which is a common requirement in calculus and advanced geometry.

Trigonometry

Trigonometry is the branch of mathematics dealing with angles, triangles, and trigonometric functions like sine, cosine, and tangent. Utilizing radians in trigonometry can offer a variety of simplifications. Many trigonometric identities and formulas assume angles are in radians, particularly within calculus applications.
Here are some reasons why radians are preferred in trigonometry:

• They simplify derivative and integral calculations involving trigonometric functions.
• Radian measure directly corresponds with the arc length of a circle.
• They make series expansion for trigonometric functions more straightforward to calculate.

When converting angles in trigonometric problems, it's crucial to ensure you're working consistent units, especially when using scientific calculators or solving within complex equations.

Mathematical Simplification

Simplification is a key concept in algebra and helps to make complex expressions more manageable and easier to interpret. During angle conversion, simplifying fractions is often necessary.
In our example of converting \(-225^{\circ}\) to radians:

• We first express it using the conversion formula as \[-225 \times \frac{\pi}{180}\]
• Simplifying fractions involves finding a common factor. Here, both 225 and 180 share a factor of 45.
• After dividing both by 45, we obtain \[\frac{-5}{4}\]

This simplification helps clarify the result \(-\frac{5}{4}\pi\), making it more concise and easier to work with in further calculations or applications.