Question

Convert each angle in radians to degrees. Express your answer in decimal form, rounded to two decimal places. 3

Short answer

171.89 degrees

Step-by-step solution

Understand the Conversion Factor

To convert an angle from radians to degrees, use the conversion factor: 1 radian = \( \frac{180}{\pi} \) degrees.

Apply the Conversion Factor

Multiply the given angle in radians by the conversion factor to convert it to degrees. For the angle 3 radians, this becomes: \( 3 \times \frac{180}{\pi} \).

Calculate the Result

Calculate the product to find the angle in degrees: \( 3 \times \frac{180}{\pi} \approx 3 \times 57.2958 \approx 171.89 \).

Key concepts

Radians to Degrees

When you work with angles in trigonometry, you will often encounter two common units: radians and degrees. Each unit serves a purpose in different mathematical contexts. Converting between these units is crucial for calculations.
Radians measure angles based on the radius of a circle. One radian describes the angle created when the arc length is equal to the circle’s radius. Degrees, however, divide a circle into 360 equal parts, making it easier for certain practical uses.
To convert an angle from radians to degrees, you need to understand the relationship between these units. This relationship is defined by the circular constant \(\frac{180}{u}\). One full revolution around a circle is 2\(\pi\) radians or 360 degrees.

Conversion Factor

The conversion factor is key to switching between radians and degrees.
As per the exercise, the conversion factor is given as:

• 1 radian = \(\frac{180}{\pi}\) degrees

To use the conversion factor, simply multiply your angle in radians by \(\frac{180}{\pi}\).
For example, to convert 3 radians to degrees, you perform the following calculation: \(3 \times \frac{180}{u}\approx 171.89\). This calculation shows that 3 radians is approximately 171.89 degrees.

Trigonometry Calculations

Understanding how to transform angles between radians and degrees is essential for trigonometry calculations. Many trigonometric functions, like sine, cosine, and tangent, rely on angle measurements. These functions often accept inputs in radians, but you might want the output in degrees.
For instance, suppose you are dealing with an angle measured in radians in a trigonometric formula. Before you can use the result practically (e.g., for construction or navigation), you might have to convert your radian measure to degrees.
This skill of converting allows you to

• accurately compare angle measures,
• understand trigonometric identities,
• and solve real-world problems involving angles.

In short, mastering this fundamental conversion enhances your efficiency when dealing with trigonometric calculations.