Question

Convert each radian measure to degrees. Express your answers as exact values and as approximate measures, to the nearest thousandth. a) \(\frac{2 \pi}{7}\) b) \(\frac{7 \pi}{13}\) c) \(\frac{2}{3}\) d) 3.66 e) -6.14 f) -20

Short answer

a) 51.429 degrees, b) 96.923 degrees, c) 38.197 degrees, d) 209.719 degrees, e) -351.929 degrees, f) -1145.915 degrees

Step-by-step solution

- Understanding the Conversion Factor

To convert from radians to degrees, use the conversion factor: \[1 \text{ radian} = \frac{180}{\pi} \text{ degrees}\]

- Convert \(\frac{2 \pi}{7}\) to Degrees

First, express the radians in terms of degrees using the conversion factor:\[\frac{2 \pi}{7} \text{ radians} \times \frac{180}{\pi} \approx 51.429 \text{ degrees}\]Exact value: \[\frac{360}{7} \text{ degrees}\] which simplifies to about 51.429 degrees.

- Convert \(\frac{7 \pi}{13}\) to Degrees

Using the conversion factor:\[\frac{7 \pi}{13} \text{ radians} \times \frac{180}{\pi} \approx 96.923 \text{ degrees}\]Exact value: \[\frac{1260}{13} \text{ degrees}\] which simplifies to about 96.923 degrees.

- Convert \(\frac{2}{3}\) to Degrees

Similarly, use the conversion factor:\[\frac{2}{3} \text{ radians} \times \frac{180}{\pi} \approx 38.197 \text{ degrees}\]Exact value: \[\frac{120}{\pi} \text{ degrees}\] which approximates to about 38.197 degrees.

- Convert 3.66 Radians to Degrees

Use the conversion factor:\[3.66 \text{ radians} \times \frac{180}{\pi} \approx 209.719 \text{ degrees}\]Exact value: \[3.66 \times \frac{180}{\pi} \text{ degrees}\]

- Convert -6.14 Radians to Degrees

Using the conversion factor:\[-6.14 \text{ radians} \times \frac{180}{\pi} \approx -351.929 \text{ degrees}\]Exact value: \[-6.14 \times \frac{180}{\pi} \text{ degrees}\]

- Convert -20 Radians to Degrees

Again, use the conversion factor:\[-20 \text{ radians} \times \frac{180}{\pi} \approx -1145.915 \text{ degrees}\]Exact value: \[-20 \times \frac{180}{\pi} \text{ degrees}\]

Key concepts

conversion factor

To convert any angle from radians to degrees, we utilize the conversion factor: \r\[ 1 \text{ radian} = \frac{180}{\text{pi}} \text{ degrees} \]\rThis means one radian is equivalent to approximately 57.2958 degrees. Essentially, you multiply any radian measure by \(\frac{180}{\pi}\) to convert it into degrees.
\r

\r
• For instance, to convert \(\frac{2 \pi}{7}\) to degrees, the calculation would look like this:
\r\[\frac{2 \pi}{7} \times \frac{180}{\pi} \]\r
• When we cancel out the \text{pi} from the numerator and denominator, it simplifies to:
\r\[ \frac{2 \times 180}{7} = \frac{360}{7}\]\r
• This calculation shows \(\frac{360}{7}\) as the exact value in degrees, which approximates to 51.429 degrees.
\r

\rBreaking it down like this should clarify how the conversion factor gets applied step by step.

exact value

An 'exact value' means a precise and accurate representation of a measurement, without any rounding or approximation. \rWhen converting radians to degrees, the exact value often includes irrational numbers like \(\pi\). \r

\r
• For example, when converting \(\frac{7 \pi}{13}\) radians, the exact value in degrees is:
\r\[ \frac{7 \pi}{13} \times \frac{180}{\pi} = \frac{7 \times 180}{13} = \frac{1260}{13} \text{ degrees}\]\r
• Hence, \(\frac{1260}{13}\) degrees represents the exact value.
\r
• This calculation avoids any approximation and retains the true measurement.
\r

\rSimilarly, for non-\(\pi\) based radians like \( \frac{2}{3} \) radians, use: \r

\r
• Exact value being
\r\[ \frac{2}{3} \times \frac{180}{\pi} \approx \frac{120}{\pi} \text{ degrees}\]\r
• The exact fraction here gives us a more precise conversion compared to its approximate form.
\r

approximate measure

When you need a more practical and easy-to-use number than an exact value, you use an approximate measure. This is especially useful when dealing with irrational numbers like \(\pi\).
\r

\r
• For example, while \(\frac{360}{7}\) is the exact value, its approximate measure could be written as:
\r52 degrees (rounded to the nearest whole number).\r
• This is essential for everyday calculations where simplicity is key.
\r

\rTo convert the value of 3.66 radians: \r

\r
• The approximate measure is calculated as:
\r\[3.66 \times \frac{180}{\pi} \approx 209.719 \text{ degrees}\]\r
• While the exact value involves more precise calculations, rounding it to the nearest thousandth, we get 209.719 degrees.
\r

\rSimilarly, for negative radians, ensure to use the same conversion techniques: \r

\r
• -6.14 radians approximates to:
\r\[ -6.14 \times \frac{180}{\pi} \approx -351.929 \text{ degrees}\]\r
• And -20 radians to:
\r\[ -20 \times \frac{180}{\pi} \approx -1145.915 \text{ degrees}\]\r
• Rounding off these values makes them useful for practical applications without losing much precision.
\r