Problem 31 - Precalculus

Question

Convert from radians to degrees. $$\frac{3 \pi}{4}$$

Step-by-Step Solution

Answer

\( \frac{3\pi}{4} \) radians is equal to \( 135 \) degrees.

Step 1: Understanding the Conversion Factor

To convert radians to degrees, we need to use the conversion factor. One full circle is equal to 360 degrees or \( 2\pi \) radians. Therefore, 1 radian equals \( \frac{180}{\pi} \) degrees.

Step 2: Applying the Conversion Factor

Multiply the radian measure \( \frac{3\pi}{4} \) by the conversion factor \( \frac{180}{\pi} \):\[\frac{3\pi}{4} \times \frac{180}{\pi}\]

Step 3: Simplifying the Expression

Simplify the expression by canceling out \( \pi \):\[\frac{3\pi \times 180}{4\pi} = \frac{3 \times 180}{4}\]Now, divide \( 540 \) by \( 4 \):\[\frac{540}{4} = 135\]

Step 4: Final Result

\( \frac{3\pi}{4} \) radians is equal to \( 135 \) degrees.

Key Concepts

Radian Measure

The concept of radian measure is essential for understanding angles in mathematics, particularly in trigonometry and calculus. A radian is a unit of angular measure used to describe the size of an angle.
One radian is the angle created when an arc length on a circle is equal to the radius of that circle. This natural unit of measurement makes radians particularly useful in mathematics due to its simplicity and relationship with the circle's radius.
Unlike degrees, which split a circle into 360 equal parts, radians break the circle into approximately 6.283 pieces, since there are about 6.283 radians in a full circle (or exactly \(2\pi\) radians).
The radian measure is compact and easy to use, especially when dealing with calculus and periodic functions, such as sine and cosine.

Conversion Factor

To convert between radians and degrees, we use a specific conversion factor. This factor is derived from the relationship between the total number of degrees in a circle (360) and the total number of radians (\(2\pi\)).

• Since \(2\pi\) radians is equivalent to 360 degrees, we can deduce that 1 radian equals \(\frac{180}{\pi}\) degrees.
• This relationship serves as our conversion factor when changing measurements from radians to degrees or vice versa.

Multiply any radian measure by this factor (\(\frac{180}{\pi}\)) to convert it into degrees.
It is a simple multiplication, making the conversion process both straightforward and reliable for various mathematical and real-world applications.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form, which often serves to make calculations easier and more manageable.
When converting radians to degrees, simplifying the expression can involve several steps: canceling out common terms, performing basic arithmetic, and ensuring that the final result is in its simplest possible form.

• For instance, when you multiply the radian measure \(\frac{3\pi}{4}\) by the conversion factor \(\frac{180}{\pi}\), the \(\pi\) terms cancel out.
• This leaves the expression \(\frac{3 \times 180}{4}\), which can be further simplified by performing division.
• The result, in this case, is 135 degrees, which is both simple and easy to understand.

Simplifying not only clears up the math but also helps ensure that the solution is easy to interpret, making it a valuable skill in any mathematical context.