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Chapter 5: Problem 56

Convert each degree measure to radian measure. Give exact answers. $$48^{\circ}$$

Short Answer

Expert verified
\( 48^{\text{\degree}} = \frac{4\text{\pi}}{15} \)

Step by step solution

01

Understand the conversion factor

To convert degrees to radians, use the conversion factor \( \frac{\text{\textpi}}{180} \). This means that \( 180^{\text{\degree}} \) is equal to \( \textpi \text{ radians} \).
02

Set up the conversion

Multiply the given degree measure by the conversion factor. For \( 48^{\text{\degree}} \), set up the conversion as follows:\[ 48^{\text{\degree}} \times \frac{\text{\pi}}{180} \]
03

Simplify the expression

Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor. Here we have: \[ \frac{48\text{\pi}}{180} = \frac{4\text{\pi}}{15} \]
04

Write the final answer

The exact radian measure for \( 48^{\text{\degree}} \) is \[ \frac{4\text{\pi}}{15} \]

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Key Concepts

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

Degree Measure
Degrees are a way to measure angles. The full circle is divided into 360 degrees.
This means an angle measuring 360 degrees completes a full circle.
A right angle is 90 degrees, so one degree is \(\frac{1}{360}\)th of a full circle.
Degrees are easy to visualize and are commonly used in everyday situations.
Radian Measure
Radians are another way to measure angles, often used in mathematics and science.
One radian is the angle created when the arc length is equal to the radius of the circle.
There are \(2\pi\) radians in a full circle because the circumference of a circle is \(2\pi r\), where \(r\) is the radius.
This gives a direct link between the angle measure and the properties of the circle itself.Radians might seem complex at first, but they simplify many mathematical formulas.
Conversion Factor
To convert from degrees to radians, use the conversion factor \(\frac{\pi}{180}\).
This factor comes from the proportional relationship between degrees and radians.
Since 180 degrees equals \pi radians, we can set up a ratio: \(\frac{180\text{ degrees}}{\pi\text{ radians}}=1\).When converting, multiply the degree measure by \(\frac{\pi}{180}\).
This shifts the angle from degrees to radians. For example, converting 48 degrees follows this format: \(48^{\circ} \times \frac{\pi}{180}\).
Simplification
Simplifying fractions is a key step in many math problems, including angle conversion.
After multiplying the degree measure by the conversion factor (\[48^{\circ} \times \frac{\pi}{180}\]), the next step is to simplify.
Find the greatest common divisor (GCD) of the numerator and the denominator.
Here, 48 and 180 are both divisible by 12. Dividing both by 12 yields: \(\frac{48\pi}{180} = \frac{4\pi}{15}\).This simplified form, \(\frac{4\pi}{15}\), is the exact radian measure that corresponds to 48 degrees. Simplification not only makes answers cleaner but often reveals deeper mathematical properties.

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Most popular questions from this chapter

Determine the period and sketch at least one cycle of the graph of each function. $$y=\tan (\pi x)$$

Find the approximate value of each expression. Round to four decimal places. $$\tan \left(-44.6^{\circ}\right)$$

A weight hanging on a vertical spring is set in motion with an upward velocity of \(4 \mathrm{cm} / \mathrm{sec}\) from its equilibrium position. Assume that the constant \(\omega\) for this particular spring and weight combination is \(\pi .\) Write the formula that gives the location of the weight in centimeters as a function of the time \(t\) in seconds. Find the period of the function and sketch its graph for \(t\) in the interval \([0,4]\).

Find the approximate value of each expression. Round to four decimal places. $$\csc \left(-44.3^{\circ}\right)$$

Graph \(y=x+\sin x\) for \(-100 \leq x \leq 100\) and \(-100 \leq y \leq 100 .\) Explain your results.
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