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Chapter 6: Problem 11

Convert to radian measure. Leave the answer in terms of \(\pi\). $$200^{\circ}$$

Short Answer

Expert verified
\(\frac{10\pi}{9}\)

Step by step solution

01

Understand the conversion factor

The conversion factor between degrees and radians is important to know. Specifically, \(180^{\circ} = \pi \text{ radians}\).
02

Set up the conversion

To convert from degrees to radians, multiply the number of degrees by the fraction \(\frac{\pi}{180}\). So, \(200^{\circ} \rightarrow 200 \times \frac{\pi}{180}\).
03

Simplify the expression

Simplify the multiplication step: \(200 \times \frac{\pi}{180} = \frac{200\pi}{180}\). Continue simplifying the fraction \(\frac{200\pi}{180} = \frac{10\pi}{9}\).
04

Write the final answer

The radian measure for \(200^{\circ}\) is \(\frac{10\pi}{9}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

degree to radian conversion
When converting degrees to radians, it's crucial to understand the relationship between these two units of angle measurement. Specifically, 180 degrees is equivalent to \(\pi\) radians. This means that any angle in degrees can be converted to radians by multiplying it by the fraction \(\frac{\pi}{180}\). For instance, to convert \(200^{\circ}\) to radians, you multiply 200 by \(\frac{\pi}{180}\). This method allows for a straightforward conversion process.
angle measurement
Angle measurement can be done in degrees or radians, and understanding both is important for various applications in mathematics and science. Degrees are commonly used in everyday scenarios, while radians are more prevalent in higher mathematics, especially in trigonometry and calculus. To put it simply:
  • 1 full circle is 360 degrees
  • 1 full circle is also \(2\pi\) radians
  • Thus, \(\pi\) radians = 180 degrees
Knowing this, one can easily convert between degrees and radians, enhancing their understanding of geometric and trigonometric concepts.
simplifying fractions
Simplifying fractions is a key skill in ensuring that mathematical expressions are presented in their simplest form. This involves reducing the numerator and denominator of the fraction to their smallest possible values while maintaining the same ratio. For example, in converting \(200^{\circ}\) to radians, we end up with \(\frac{200\pi}{180}\). To simplify this:
  • Find the greatest common divisor (GCD) of 200 and 180, which is 20.
  • Divide both the numerator and the denominator by their GCD: \(\frac{200 \pi}{180} = \frac{200/20 \pi}{180/20} = \frac{10 \pi}{9}\).
Thus, \(\frac{200\pi}{180}\) simplifies to \(\frac{10\pi}{9}\). This not only makes the fraction easier to work with but also ensures accuracy in further calculations.

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Most popular questions from this chapter

Find the reference angle and the exact function value if they exist. $$\csc 225^{\circ}$$

For angles of the following measures, state in which quadrant the terminal side lies. It helps to sketch the angle in standard position. $$315^{\circ}$$

Make a hand-drawn graph of the function. Then check your work using a graphing calculator. $$g(x)=\log _{2} x$$

Distance Between Towns. From a hot-air balloon \(2 \mathrm{km}\) high, the angles of depression to two towns in line with the balloon and on the same side of the balloon are \(81.2^{\circ}\) and \(13.5^{\circ} .\) How far apart are the towns?

To find the distance between two points on the earth when their latitude and longitude are known, we can use a right triangle for an excellent approximation if the points are not too far apart. Point \(A\) is at latitude \(38^{\circ} 27^{\prime} 30^{\prime \prime} \mathrm{N},\) longitude \(82^{\circ} 57^{\prime} 15^{\prime \prime} \mathrm{W},\) and point \(B\) is at latitude \(38^{\circ} 28^{\prime} 45^{\prime \prime} \mathrm{N},\) longitude \(82^{\circ} 56^{\prime} 30^{\prime \prime} \mathrm{W}\). Find the distance from \(A\) to \(B\) in nautical miles. (One minute of latitude is one nautical mile.)
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