Problem 98 - Algebra and Trigonometry

Question

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Convert \(\frac{7 \pi}{3}\) radians to degrees.

Step-by-Step Solution

Answer

420 degrees

Step 1 - Understand the Conversion Factor

To convert radians to degrees, use the conversion factor where 1 radian equals \( \frac{180}{\pi} \) degrees.

Step 2 - Multiply by Conversion Factor

Multiply the given radian measure \( \frac{7 \pi}{3} \) by the conversion factor \( \frac{180}{\pi} \) degrees per radian. \( \frac{7 \pi}{3} \times \frac{180}{\pi} \)

Step 3 - Simplify the Expression

Next, simplify the expression by canceling out \pi: \( \displaystyle \frac{7 \cdot 180}{3} \).

Step 4 - Calculate the Final Value

Finally, perform the multiplication and division: \( \frac{7 \cdot 180}{3} = 7 \cdot 60 = 420 \).

Step 5 - State the Answer

The final answer is \( 420 \) degrees.

Key Concepts

radian measure

A radian is a unit of measurement for angles. It comes from the radius of a circle. Imagine if you took the radius of a circle and 'wrapped' it around the circle's circumference. The angle formed at the circle's center is 1 radian. There are 2π (about 6.28) radians in a full circle. Understanding radians is crucial in more advanced math, especially calculus, because it simplifies complex equations and calculations.

degree measure

Degrees are another way to measure angles. Unlike radians, degrees are more commonly used in everyday situations. A full circle is 360 degrees. This system is rooted in historical and practical use. For example, dividing a circle into 360 parts made calculating with whole numbers easier. For many students, degrees feel more intuitive. However, understanding both radians and degrees is essential, particularly in fields like trigonometry and physics.

conversion factor

To convert between radians and degrees, use the conversion factor. The key relationship here is that 1 radian equals \(\frac{180}{\pi}\) degrees. This comes from the full circle equivalences: 2π radians equal 360 degrees. To convert from radians to degrees, multiply by \(\frac{180}{\pi}\): \(\text{{radians}} \times \frac{180}{\pi} \). Conversely, to convert degrees to radians, multiply by \(\frac{\pi}{180}\). This factor helps smoothly transition between the two measurement systems in mathematical problems.

trigonometry

Trigonometry is the branch of mathematics dealing with angles and their relationships. It's especially key in understanding circles, waves, and oscillations. Radians are often favored in trigonometry because they simplify derivatives and integrals. Typical trigonometric functions like sine, cosine, and tangent become more straightforward when angles are measured in radians. Having a solid grasp of both radians and degrees empowers you to solve a wide range of problems, from simple angle conversions to more complex wave functions.