Problem 20 - Basic Technical Mathematics with Calculus

Question

Change the given angles in radians to equal angles expressed in degrees to the nearest \(0.01^{\circ}\) $$-3.642 \mathrm{rad}$$

Step-by-Step Solution

Answer

-3.642 radians is approximately -208.73°.

Step 1: Understand the relationship between radians and degrees

One full circle is equal to \(2\pi\) radians or \(360^{\circ}\). This means \(\pi\) radians is equivalent to \(180^{\circ}\). We can use this relationship to convert from radians to degrees.

Step 2: Set up the conversion formula

To convert radians to degrees, use the formula: \[\text{Degrees} = \text{Radians} \times \left(\frac{180^{\circ}}{\pi}\right)\] This formula is based on the fact that \(\pi\) radians equal \(180^{\circ}\).

Step 3: Substitute the given value into the formula

Substitute \(-3.642\) for the radian value in the conversion formula: \[\text{Degrees} = -3.642 \times \left(\frac{180^{\circ}}{\pi}\right)\]

Step 4: Calculate the degrees

Perform the calculation: \[\text{Degrees} = -3.642 \times \left(\frac{180}{3.1416}\right) \approx -208.73^{\circ}\] Use \(\pi \approx 3.1416\) for calculation to achieve the precision needed.

Step 5: Round to nearest hundredth of a degree

The result from the calculation is already rounded to two decimal places. The angle \(-3.642\,\text{rad}\) is approximately \(-208.73^{\circ}\) to the nearest hundredth of a degree.

Key Concepts

Angle Measurement

Angles are fundamental components in geometry and trigonometry. They help us understand rotation and direction.
In mathematics, angles can be measured in different units including degrees and radians.The degree is a unit of measurement where a full circle is divided into 360 equal parts. Each part is one degree. Radians provide another way to measure angles, based on the radius of a circle.
One full circle measures as approximately 6.283 radians or exactly 2\(\pi\) radians.

Trigonometry

Trigonometry is a branch of mathematics that studies relationships involving lengths and angles in triangles.
It's widely used in various fields like engineering, physics, and astronomy. Converting between radians and degrees is essential in trigonometry.
Since many trigonometric functions, like sine and cosine, require angle inputs, understanding these units ensures precise calculations in problems related to periodic phenomena.

Mathematics Education

Learning about angle measurements in both degrees and radians is a key part of mathematics education.
Understanding these concepts lays the foundation for more advanced studies in mathematics and sciences. Educators often emphasize the importance of converting between these units. Once you master this conversion, you'll find it easier to solve a wide range of math problems.
It’s critical to practice these conversions to gain a deeper understanding and fluency in mathematical reasoning.

Degree Calculation

To convert radians to degrees, employ the conversion factor \(\frac{180^{\circ}}{\pi}\).
This comes from the equivalence where \(\pi\) radians equal \(180^{\circ}\). Here's a simple process:

• Multiply the number of radians by \(\frac{180}{\pi}\).
• Perform the multiplication to get the result in degrees.
• For precise calculations, round off to the nearest hundredth.

By following these steps, the radian value \(-3.642\) converts to approximately \(-208.73^{\circ}\). Understanding and mastering this process enhances precision in scientific and engineering applications.