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Magazine Chapter 9: Problem 87
Convert each radian measure to degrees. Round answers to the nearest minute. $$-1.3$$
Short Answer
Expert verified
-1.3 radians is approximately -74掳 29'.
Step by step solution
01
Understand Radian to Degree Conversion
The conversion between radians and degrees is based on the fact that \(2\pi\) radians is equivalent to \(360^\circ\). Therefore, to convert radians to degrees, we use the formula: \[\text{Degrees} = \text{Radians} \times \left(\frac{180^\circ}{\pi}\right)\] This formula lets you convert any radian measure into degrees by multiplying the radian value by \frac{180}{\pi}.
02
Apply the Conversion Formula
We need to convert \(-1.3\) radians to degrees using the formula: \[ \text{Degrees} = -1.3 \times \left(\frac{180^\circ}{\pi}\right) \] Calculate the value to find the degree measure.
03
Calculate the Degree Value
Perform the multiplication: \[ \text{Degrees} = -1.3 \times 57.2958 \approx -74.4846^\circ \] Thus, \(-1.3\) radians is approximately \(-74.4846^\circ\).
04
Convert Decimal Degrees to Degrees and Minutes
The degree value \(-74.4846^\circ\) needs to be converted to degrees and minutes. Take the decimal part \(.4846\), and multiply it by \(60\) (since there are 60 minutes in a degree) to find the minutes: \[0.4846 \times 60 \approx 29\] Thus, \-74.4846^\circ\ is approximately \-74^\circ 29'\.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Radians
Radians are a way to measure angles, just like degrees. They are often used in mathematics, especially in trigonometry and calculus.
One radian is defined based on the radius of a circle. Imagine a circle with a certain radius. If you take that radius and "wrap" it around the circle, the angle you create at the center is one radian.
In simpler terms:
- A radian measures the angle based on the arc created by the radius of a circle.
- It is often used because it simplifies many equations in math, particularly in calculus.
- One full circle is equal to 2蟺 radians.
Degrees
Degrees are the more familiar unit for measuring angles, especially in fields like geometry and everyday life.
Degrees divide a full circle into 360 equal parts. Each part is one degree, which gives us the unit's name.
A few key points about degrees:
- A full rotation around a circle equals 360 degrees.
- Commonly used in navigation, geography, and when giving simple, everyday descriptions of angles.
- Useful for breaking down angles into smaller parts like minutes and seconds for precision.
Conversion Formula
Converting between radians and degrees is essential and can be done using a straightforward formula. Since a full circle measures both 360 degrees and 2蟺 radians, we use this relationship:When converting from radians to degrees:
- Use the formula: \ \[ \text{Degrees} = \text{Radians} \times \left(\frac{180^\circ}{\pi}\right) \]
- This formula multiplies the radian measure by \(\frac{180}{\pi}\) to get the equivalent degree measure.
- For example, \(-1.3\) radians can be converted with this formula to approximately \(-74.4846^\circ\).
Decimal Degrees
Decimal degrees reflect an angle's measurement as a decimal number, providing a precise value. However, sometimes angles need breaking down further into degrees and minutes.The process involves:
- Identifying the whole number part as the degree measure.
- Multiplying the decimal part by 60 to determine the minutes, since there are 60 minutes in one degree.
- Adding these results together gives you the full angle measure in degrees and minutes.
