Problem 78 - Algebra and Trigonometry Real Mathematics, Real People

Question

Convert the angle measure from radians to degrees. Round your answer to three decimal places. $$\frac{5 \pi}{11}$$

Step-by-Step Solution

Answer

The equivalent angle measure in degrees is \(81.818\) degrees.

Step 1: Identify the Given Angle Measure in Radians

The given angle measure in radians is \(\frac{5 \pi}{11}\).

Step 2: Apply the Conversion Factor

To convert from radians to degrees, use the conversion factor: \(\frac{180}{\pi}\). So we will multiply our radians measure by this factor: \(\frac{5 \pi}{11} \times \frac{180}{\pi}\).

Step 3: Perform the Calculations

Perform the calculations: the \(\pi\) terms cancel out, leaving \(\frac{5 \times 180}{11} = 81.81818\).

Step 4: Round to Three Decimal Places

Since our final answer needs to be rounded to three decimal places, we obtain \(81.818\).

Key Concepts

Radians to Degrees

When converting an angle from radians to degrees, it's essential to understand the relationship between these two units of measurement. **Radians** and **degrees** are both units used to measure angles. While a full circle is 360 degrees, the same circle is equivalent to \(2\pi\) radians. The conversion between these units is crucial in mathematics, especially in trigonometry and calculus.

To convert radians to degrees, the formula used is:

• Multiply the radian measure by \(\frac{180}{\pi}\).

This factor, \(\frac{180}{\pi}\), is derived from the equivalence of \(\pi\) radians to 180 degrees. This conversion allows us to express an angle measured in radians as degrees, providing a more intuitive understanding for many practical applications.

In our exercise, we converted \(\frac{5 \pi}{11}\) radians to degrees which resulted in approximately 81.818 when rounded to three decimal places.

Radian Measure

Understanding the concept of radians is vital for angle measurement. Unlike degrees, which may seem more intuitive and familiar from daily life contexts, radians provide a more natural measure in mathematics, particularly when dealing with periodic functions and circular motion.

A **radian** is defined using the radius of a circle. When an arc's length is equal to the radius of the circle, the angle subtended at the circle's center is one radian. Thus, one full revolution (a complete circle) is \(2\pi\) radians because the circumference of a circle is \(2\pi\) times the radius.

Radian measure simplifies many mathematical calculations, allowing for direct use in calculus and trigonometric identities. When working with functions such as sine, cosine, and tangent, radians offer a more streamlined approach to derivatives and integrals. This distinctive measure is deeply embedded into higher-level mathematics.

Conversion Factor

A **conversion factor** acts as a bridge between different units of measurement. In angle conversions between radians and degrees, the conversion factor is \(\frac{180}{\pi}\). This specific factor comes from the ratio of the full circle measurements: 360 degrees divided by \(2\pi\) radians.

Why use a conversion factor? It standardizes conversion processes, ensuring accuracy and consistency across calculations. By multiplying a radian measure by \(\frac{180}{\pi}\), we translate it into a unit (degrees) that many people find more accessible and easier to relate to in everyday experiences.

In calculations, such as in our exercise, this factor effectively and accurately converts \(\frac{5 \pi}{11}\) radians to approximately 81.818 degrees. Maintaining precision in these conversions is essential, particularly when participating in fields requiring exact specifications, like engineering or physics.