Problem 10 - Algebra 2

Question

Write each measure in degrees. Round your answer to the nearest degree, if necessary. \(-3\) radians

Step-by-Step Solution

Answer

The given measure of \(-3\) radians is approximately equivalent to \(-172\) degrees.

Step 1: Identify the given

The given measure in radians is -3. The task is to convert this measurement to degrees.

Step 2: Apply the conversion factor

To convert from radians to degrees, multiply the given radian measure by \(\frac{180}{\pi}\).

Step 3: Call to compute operation

So, the equivalent in degrees will be \(-3 \times \frac{180}{\pi}\).

Step 4: Perform the calculation

Carrying out the multiplication, we get approximately -172 degrees.

Key Concepts

Radians

Radians are a way of measuring angles, much like degrees. But instead of breaking a circle into 360 parts, radians relate directly to the circle's radius. One full circle, or 360 degrees, is equivalent to approximately 6.2832 radians, which is the same as 2\(\pi\) radians.

• 1 radian is the angle created when the arc length equals the radius.
• Because 2\(\pi\) radians equals 360 degrees, 1 radian is roughly 57.2958 degrees.

Understanding radians is important when working with circles and mathematical functions such as trigonometry. It allows for seamless integration between geometry and calculus due to its natural fit within the concept of circles.

Conversion Factor

A conversion factor is a ratio used to change units from one system to another. In this case, to convert from radians to degrees, you use the conversion factor \(\frac{180}{\pi}\).

• The factor \(\frac{180}{\pi}\) translates radians to degrees.
• This is because there are 180 degrees in half a circle (or \(\pi\) radians).

Using this conversion factor helps whenever you need to move between the two measuring systems. Memorizing this simple fraction can greatly assist when doing either quick computations or more complex calculations.

Multiplication

To convert radian measures to degrees, it involves a basic multiplication process. Just take the number in radians and multiply it by the conversion factor \(\frac{180}{\pi}\). This operation will give the measure in degrees. Let's apply this multiplication process:- For the given example, - Multiply -3 (radians) by \(\frac{180}{\pi}\) to convert to degrees. Mathematically, it's expressed as:\[-3 \times \frac{180}{\pi} \approx -171.887\]This step is critical as it directly turns the radian measure into degrees and helps you interpret or use the angle in more familiar terms.

Rounding

Rounding, in mathematics, is the process of making a number simpler while keeping its value close to what it was originally. After conversions, you may get decimal results that need rounding to make them more manageable or readable.

• In our conversion example, -171.887 degrees is rounded to -172 degrees.
• We round to the nearest whole number for simplicity, especially when required by the exercise or context in which the number will be used.

Rounding can also help minimize errors when numbers are used in subsequent calculations, especially in real-world measurements where whole numbers are preferable.