Problem 35 - Algebra and Trigonometry

Question

In Exercises \(35-40,\) convert each angle in radians to degrees. Round to two decimal places. 2 radians

Step-by-Step Solution

Answer

2 radians is equal to approximately 114.59 degrees.

Step 1: Determine the conversion factor

The conversion factor between radians and degrees is approximately 57.296. This is because 1 complete rotation in radians is \(2\pi\), which is equal to \(360^{\circ}\). Thus, \(1 \text{ radian} = \frac{360^{\circ}}{2*pi} \approx 57.296^{\circ}\).

Step 2: Convert radians to degrees

Using the conversion factor from Step 1, you can convert 2 radians to degrees. Multiply the given radian measure by the conversion factor, \(2 \text{ rad} * 57.296^{\circ}/\text{rad} = 114.59^{\circ}\).

Key Concepts

Angle Measurement

Angles are a fundamental aspect of geometry, which involve measuring the rotation between two intersecting lines or rays. This measurement can be performed in different ways, but the most common are degrees and radians.

Understanding angle measurement is essential for solving problems, as angles describe the amount of turn and are crucial in various fields such as mathematics, physics, and engineering. In angle measurement:

• A full circle rotation is equivalent to a total of 360 degrees or approximately 6.283 radians (since there are \(2\pi\) radians in a full circle).
• Commonly seen angles like right angles measure 90 degrees or \(\frac{\pi}{2}\) radians.
• Angles can also be represented as fractions or multiples of these full rotations.

This duality in measurement systems requires conversions between degrees and radians for consistency and precision in calculations.

Degrees

Degrees are perhaps the most familiar unit for measuring angles, often used in everyday concepts like time or navigation. The degree is a unit that divides a complete circle into 360 equal parts. Each small part is one degree.

The degree system is intuitive because of its divisibility; for example:

• An acute angle is less than 90 degrees.
• A right angle equals exactly 90 degrees.
• An obtuse angle is more than 90 degrees but less than 180 degrees.

Understanding degrees helps in visualizing and calculating angles in practical situations. Though it is commonly used in daily life instead of radians, certain mathematical calculations benefit from using radians.

Conversion Factor

A conversion factor is a mathematical tool that helps translate one unit of measurement to another. In converting angles from radians to degrees, the conversion factor is crucial. This factor is derived from the relationship between the units:

- We know that a full circle is \(360^{\circ}\) and also \(2\pi\) radians.- Thus, dividing \(360^{\circ}\) by \(2\pi\) gives us the conversion factor, which approximates to 57.296 degrees per radian.Using this conversion factor:

• Multiply the radian value by 57.296 to get its equivalent in degrees.
• For example, converting 2 radians: \(2 \times 57.296 = 114.592^{\circ}\).
• The conversion factor simplifies calculation between the two angle measurements.

This factor facilitates smooth transitions between different systems, ensuring accuracy in mathematical work.