Problem 26 - Algebra 2

Question

Rewrite each degree measure in radians and each radian measure in degrees. \(-\frac{\pi}{4}\)

Step-by-Step Solution

Answer

\(-\frac{\pi}{4}\) radians is \(-45\) degrees.

Step 1: Convert radians to degrees

To convert a radian measure to degrees, use the conversion \[1 ext{ radian} = \frac{180}{\pi} ext{ degrees}\]So, multiply \(-\frac{\pi}{4}\) by \(\frac{180}{\pi}\) to convert it into degrees.

Step 2: Perform the coefficient multiplication

The expression now becomes:\[-\frac{\pi}{4} \times \frac{180}{\pi} = -\frac{\pi \times 180}{4 \times \pi}\]Cancel out the \(\pi\) in the numerator and the denominator.

Step 3: Simplify the expression

After canceling \(\pi\), the expression is simplified to:\[-\frac{180}{4}\]Calculate \(-\frac{180}{4}\) to find the degree measure.

Step 4: Calculate the final degree measure

Divide 180 by 4:\[-\frac{180}{4} = -45\]Thus, \(-\frac{\pi}{4}\) radians is equal to \(-45\) degrees.

Key Concepts

Radians to Degrees

To convert a measure from radians to degrees, you need to use a specific conversion factor. This conversion factor is based on the relationship between radians and degrees in a circle. One full circle is 360 degrees or \(2\pi\) radians.

Therefore, we can establish that:\[1 \text{ radian} = \frac{180}{\pi} \text{ degrees}\]

When converting, simply multiply the radian measure by \(\frac{180}{\pi}\). In this case, for \(-\frac{\pi}{4}\), the conversion would look like this:

- Multiply \(-\frac{\pi}{4}\) by \(\frac{180}{\pi}\).
- The \(\pi\) cancels out, allowing you to easily solve for the degree measure.

Using this conversion technique ensures that all radian measures can be accurately transformed into degrees, allowing you to interpret angles in different formats with ease.

Degree Measure

Understanding degree measure is crucial when dealing with angles. Degrees quantify the amount of rotation from the starting point on a circle's circumference.

For instance, a 180-degree angle is a straight line. It takes up half of the circular path. Visualizing degrees helps in comprehending how angles appear and change in spacial terms.

When converting an angle from radians, this degree measure gives you a clearer, more intuitive view of the angle's size. For example, converting \(-\frac{\pi}{4}\) radians ends up equating to \(-45\) degrees, which is easily recognizable as less than a right angle and rotating backwards on the circle's path.

This insight into degree measure allows you to make quick comparisons and judgements in various mathematics and real-world applications.

Coefficient Multiplication

Once you've set up the conversion factor for turning radians into degrees, it's time to perform coefficient multiplication. This might sound complicated, but it's simply about multiplying numbers and managing the coefficients in fractions.

For the conversion \(-\frac{\pi}{4} \times \frac{180}{\pi}\), you treat each part separately:

- Multiply \(-\pi\) by 180
- Handle the denominator as a separate entity which is \(4\pi\)

Because \(\pi\) appears in both the numerator and denominator, they cancel each other out. Doing what is called simplifying the coefficients. This step reduces the fraction and makes the problem significantly easier to solve. Coefficient multiplication is therefore an essential tactic for simplifying mathematical expressions and reaching a solution efficiently.

Simplifying Expressions

Simplifying expressions is a foundational skill necessary for solving most mathematical problems. After multiplying and canceling out like-coefficients, as shown in our step-by-step solution, you often end up with a simpler fraction that needs one last calculation.

In our example, post-simplification, we derive \(-\frac{180}{4}\). To simplify this expression, you simply divide the top number, the numerator, by the bottom number, the denominator:

- Perform \(-180 \div 4 = -45\)

This final calculation gives you the answer in degrees, \(-45\). Simplifying, in this way, ensures you cut down on unnecessary complexity and make space for easier, more direct problem-solving. It is an essential step in accurately resolving mathematical conversions and achieving concise results.