Problem 36 - Algebra and Trigonometry

Question

In Exercises \(35-40,\) convert each angle in radians to degrees. Round to two decimal places. 3 radians

Step-by-Step Solution

Answer

The angle in degrees is approximately 171.89.

Step 1: Understand the conversion factor

Identify that the conversion factor from radians to degrees is \( \frac{180}{\pi} \). This is a known mathematical constant.

Step 2: Multiply by the conversion factor

The angle given 3 radians, must be converted to degrees. This is achieved by multiplying the radians by the conversion factor, i.e., \(3 \times \frac{180}{\pi}\)

Step 3: Perform the calculation

Calculating \(3 \times \frac{180}{\pi}\) yields approximately 171.89 degrees. This value can be further rounded according to the problem's instruction.

Key Concepts

Angle Measurement

When we talk about angle measurement, we're referencing two main systems: degrees and radians. Understanding these systems allows us to measure and describe angles effectively.

**Degrees** are more commonly used in daily life. Picture a full circle; it holds 360 degrees. So, one degree is a tiny slice of that circle.

**Radians** are another way to measure them, often used in mathematics, especially in calculus and trigonometry. One radian is the angle you get when the arc length equals the radius of the circle.

To connect radians with degrees:

• A full circle in radians is often denoted as 2π radians.
• In degrees, that same full circle is 360 degrees.

This means that 2π radians equals 360 degrees. From this, we extract our conversion factor to switch between degrees and radians.

Mathematical Constant

A mathematical constant is a special number that holds a fixed value no matter where or how it's used. In the case of angle conversion, the constant \(\pi\) (pi) plays a crucial role.

**Pi (\( \pi \)**

• Pi is approximately equal to 3.14159.
• It's the ratio of a circle's circumference to its diameter.

This constant allows us to switch smoothly between radians and degrees. The conversion factor, \(\frac{180}{\pi}\), arises naturally when considering the relationship that the complete angle of a circle is 2π radians or 360 degrees.

Using this factor, any angle in radians can be translated to degrees by multiplying the radians with \(\frac{180}{\pi}\). This relationship is essential in mathematics whenever we need to change the unit of angles.

Rounding Numbers

Rounding numbers is the process of adjusting a number to make it simpler or easier to work with, while keeping it close to what it was originally. This practice is very common when dealing with angles after conversion from radians to degrees.

**Why Round Numbers?**

• Simplifies complex numbers.
• Makes results easy to understand and communicate.
• Helps maintain accuracy within a required degree of precision.

In our example problem, we ended up with an angle of 171.8854 degrees after conversion. Given the instructions to round to two decimal places:

• Look at the third decimal place, which is 5.
• Since 5 is greater than or equal to 5, we increase the second decimal place by one, leading to 171.89.

This simple step ensures we maintain a balance between the precision of the calculation and its readability, matching the requirements of many math problems.