Chapter 4: Problem 4
Convert each radian measure to degrees. Express your answers as exact values and as approximate measures, to the nearest tenth of a degree, if necessary. a) \(\frac{\pi}{6}\) b) \(\frac{2 \pi}{3}\) c) \(-\frac{3 \pi}{8}\) d) \(-\frac{5 \pi}{2}\) e) 1 f) 2.75
Short Answer
Expert verified
a) 30° b) 120° c) -67.5° d) -450° e) 57.3° f) 157.5°
Step by step solution
01
Conversion formula
To convert radians to degrees, use the formula: \[\text{Degrees} = \text{Radians} \times \frac{180}{\text{π}}\]
02
Convert \( \frac{\pi}{6} \)
For \( \frac{\pi}{6} \):\[\text{Degrees} = \frac{\pi}{6} \times \frac{180}{\text{π}} = 30^\text{°}\]The exact value is 30°. Approximated to the nearest tenth, it's still 30.0°.
03
Convert \( \frac{2 \pi}{3} \)
For \( \frac{2 \pi}{3} \):\[\text{Degrees} = \frac{2 \pi}{3} \times \frac{180}{\text{π}} = 120^\text{°}\]The exact value is 120°. Approximated to the nearest tenth, it's still 120.0°.
04
Convert \( -\frac{3 \pi}{8} \)
For \( -\frac{3 \pi}{8} \):\[\text{Degrees} = -\frac{3 \pi}{8} \times \frac{180}{\text{π}} = -67.5^\text{°} \]The exact value is -67.5°. Approximated to the nearest tenth, it's still -67.5°.
05
Convert \( -\frac{5 \pi}{2} \)
For \( -\frac{5 \pi}{2} \):\[\text{Degrees} = -\frac{5 \pi}{2} \times \frac{180}{\text{π}} = -450^\text{°}\]The exact value is -450°. Approximated to the nearest tenth, it's still -450.0°.
06
Convert 1 radian
For 1 radian:\[\text{Degrees} = 1 \times \frac{180}{\text{π}} \approx 57.3^\text{°}\]The approximation to the nearest tenth of a degree is 57.3°.
07
Convert 2.75 radians
For 2.75 radians:\[\text{Degrees} = 2.75 \times \frac{180}{\text{π}} \approx 157.5^\text{°}\]The approximation to the nearest tenth of a degree is 157.5°.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Angle Conversion
When dealing with angles, conversions between different units of measurement can be very useful. The two most common units for measuring angles are degrees and radians. To convert an angle from radians to degrees, you can use the following formula:
\[\text{Degrees} = \text{Radians} \times \frac{180}{\text{π}}\]
This formula is derived from the fact that a full circle is 360 degrees, which is also equal to 2π radians.
Understanding how to convert between radians and degrees is crucial because some situations or fields might prefer one measurement over the other. For example, most mathematical problems and academic exercises might use radians, while everyday applications like navigation or area calculations might use degrees.
This knowledge is not only academically beneficial but also practically useful.
\[\text{Degrees} = \text{Radians} \times \frac{180}{\text{π}}\]
This formula is derived from the fact that a full circle is 360 degrees, which is also equal to 2π radians.
Understanding how to convert between radians and degrees is crucial because some situations or fields might prefer one measurement over the other. For example, most mathematical problems and academic exercises might use radians, while everyday applications like navigation or area calculations might use degrees.
This knowledge is not only academically beneficial but also practically useful.
Radian Measure
A radian is a unit of angle measurement that is based on the radius of a circle. One radian is defined as the angle you get when you take the radius of a circle and lay it along the circle's circumference.
Mathematically, the relationship between the circumference of a circle and its radius is given by:
\[ \text{Circumference} = 2 \times \text{π} \times \text{Radius} \]
Because of this relationship, there are 2π radians in a full circle (360 degrees). This makes radians a very natural unit for measuring angles in terms of proportions of the circle.
In various fields such as calculus and trigonometry, radians are the preferred unit of measurement because they provide a more straightforward and intuitive way to understand and derive mathematical concepts.
They are especially useful when dealing with periodic functions like sine and cosine, where the angle's value often needs to be expressed in terms of π.
Mathematically, the relationship between the circumference of a circle and its radius is given by:
\[ \text{Circumference} = 2 \times \text{π} \times \text{Radius} \]
Because of this relationship, there are 2π radians in a full circle (360 degrees). This makes radians a very natural unit for measuring angles in terms of proportions of the circle.
In various fields such as calculus and trigonometry, radians are the preferred unit of measurement because they provide a more straightforward and intuitive way to understand and derive mathematical concepts.
They are especially useful when dealing with periodic functions like sine and cosine, where the angle's value often needs to be expressed in terms of π.
Degrees
Degrees are another unit for measuring angles. Unlike radians, degrees divide a full circle into 360 equal parts.
Each degree is \[ \frac{1}{360} \] of a full circle. This division is believed to have originated from ancient astronomers who divided the day into 360 parts, taking into account yearly changes in the sky.
Degrees are widely used in everyday contexts. For instance, weather forecasts, engineering, and construction make extensive use of degrees for their simplicity and ease of understanding.
When working with degrees, it's important to remember some common benchmark angles:
However, always remember that degrees can be converted back to radians if necessary using the inverse of the radian-to-degree formula: \[ \text{Radians} = \text{Degrees} \times \frac{\text{π}}{180} \].
Each degree is \[ \frac{1}{360} \] of a full circle. This division is believed to have originated from ancient astronomers who divided the day into 360 parts, taking into account yearly changes in the sky.
Degrees are widely used in everyday contexts. For instance, weather forecasts, engineering, and construction make extensive use of degrees for their simplicity and ease of understanding.
When working with degrees, it's important to remember some common benchmark angles:
- 90 degrees (a right angle)
- 180 degrees (a straight line)
- 360 degrees (a full circle)
However, always remember that degrees can be converted back to radians if necessary using the inverse of the radian-to-degree formula: \[ \text{Radians} = \text{Degrees} \times \frac{\text{π}}{180} \].