Problem 114
Question
Convert each angle in degrees to radians. Write the answer as a multiple of \(\pi\). $$15^{\circ}$$
Step-by-Step Solution
Verified Answer
\( \frac{\pi}{12} \).
1Step 1: Understand the Conversion Factor
To convert degrees to radians, we use the conversion factor \( \frac{\pi}{180} \). This is because \( 180^{\circ} \) is equal to \( \pi \) radians.
2Step 2: Apply the Conversion Factor
Multiply the given angle in degrees, \( 15^{\circ} \), by the conversion factor \( \frac{\pi}{180} \). This gives us: \[15^{\circ} \times \frac{\pi}{180} = \frac{15\pi}{180}\]
3Step 3: Simplify the Fraction
Now, we simplify the fraction \( \frac{15\pi}{180} \). Both 15 and 180 are divisible by 15. So, \( \frac{15}{180} = \frac{1}{12} \). This makes our expression: \[ \frac{\pi}{12}\]
4Step 4: Write the Final Answer
The angle \( 15^{\circ} \) is equal to \( \frac{\pi}{12} \) radians.
Key Concepts
Degrees to RadiansConversion FactorSimplifying FractionsRadian Measure
Degrees to Radians
Converting an angle from degrees to radians involves a simple mathematical process. Degrees are a unit of angular measurement used widely in everyday life. However, in mathematics and particularly in calculus, radians provide a more natural measure.
Radians are based on the radius of a circle. When you have an angle measured in degrees, like \(15^{\circ}\), you are often required to convert it to radians. This conversion is important because many mathematical formulas, like those involving trigonometric functions, are more straightforward when using radians.
The conversion process itself relies on the fact that a full circle is \(360^{\circ}\) and is equivalent to \(2\pi\) radians. Hence, \(180^{\circ}\) equals \(\pi\) radians.
Radians are based on the radius of a circle. When you have an angle measured in degrees, like \(15^{\circ}\), you are often required to convert it to radians. This conversion is important because many mathematical formulas, like those involving trigonometric functions, are more straightforward when using radians.
The conversion process itself relies on the fact that a full circle is \(360^{\circ}\) and is equivalent to \(2\pi\) radians. Hence, \(180^{\circ}\) equals \(\pi\) radians.
Conversion Factor
To switch from degrees to radians, we use a specific conversion factor: \(\frac{\pi}{180}\). This factor is derived from knowing that \(180^{\circ} = \pi\) radians.
This means every degree is equivalent to \(\frac{\pi}{180}\) radians. By multiplying the number of degrees by this factor, we convert the angle to radians.
The idea is straightforward: multiply the degree measurement by \(\frac{\pi}{180}\) to find its radian equivalent. By understanding this formula, converting any degree measurement to radians becomes a breeze.
This means every degree is equivalent to \(\frac{\pi}{180}\) radians. By multiplying the number of degrees by this factor, we convert the angle to radians.
- For example, starting with \(15^{\circ}\), you apply the factor by:
\[ 15 \times \frac{\pi}{180} \]
The idea is straightforward: multiply the degree measurement by \(\frac{\pi}{180}\) to find its radian equivalent. By understanding this formula, converting any degree measurement to radians becomes a breeze.
Simplifying Fractions
Once you have applied the conversion factor and obtained a fraction of \(\pi\), the next step is simplification. Simplifying a fraction involves reducing it to its lowest terms.
For the example of \( \frac{15\pi}{180}\), both 15 and 180 can be divided by 15, the greatest common divisor. Simplifying:
So, \( \frac{15\pi}{180} \) simplifies to \( \frac{\pi}{12} \).
This is a crucial step in the conversion process, as it ensures the radian measure is presented in the simplest and most understandable form.
For the example of \( \frac{15\pi}{180}\), both 15 and 180 can be divided by 15, the greatest common divisor. Simplifying:
- 15 divided by 15 is 1
- 180 divided by 15 is 12
So, \( \frac{15\pi}{180} \) simplifies to \( \frac{\pi}{12} \).
This is a crucial step in the conversion process, as it ensures the radian measure is presented in the simplest and most understandable form.
Radian Measure
After conversion and simplification, we arrive at the radian measure. In our example, \( 15^{\circ} \) is equal to \( \frac{\pi}{12} \) radians.
Radians offer an elegant way to describe angles because they relate directly to the arc length of a circle. One radian represents the angle made when the arc length is equal to the radius of the circle.
This property of radians makes them highly useful in calculus and trigonometric functions, where they often simplify the expressions and equations you work with.
By understanding the radian measure, students are better equipped to tackle problems across various fields of science and engineering.
Radians offer an elegant way to describe angles because they relate directly to the arc length of a circle. One radian represents the angle made when the arc length is equal to the radius of the circle.
This property of radians makes them highly useful in calculus and trigonometric functions, where they often simplify the expressions and equations you work with.
- For practical angle measurements in engineering, physics, and mathematics, radians are more universally adopted.
By understanding the radian measure, students are better equipped to tackle problems across various fields of science and engineering.
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