Problem 118 - Algebra and Trigonometry Real Mathematics, Real People

Question

Convert the angle measure from degrees to radians. Round your answer to three decimal places. $$-0.46^{\circ}$$

Step-by-Step Solution

Answer

The angle measure in radians is \(-0.008\) radians

Step 1 Convert the degree measurement to radian

To convert the angle from degrees to radians, the formula: radian measure = degree measure × \(\frac{\pi}{180}\) is used. Now, substituting the degree measure into the formula gives us the following calculation. \(-0.46 * \frac{\pi}{180}\).

Step 2 Calculate using rounding of answer

Performing the multiplication and rounding off the answer to three decimal places using the rules of significant figures, we get the radian measurement

Key Concepts

Angle Conversion

Converting angles from degrees to radians is a fundamental task in mathematics, particularly in trigonometry and various fields of science and engineering. This process is rooted in the definition of a radian: one radian is the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle.

In practice, to convert an angle from degrees to radians, you use the relationship that 180 degrees is equivalent to \(\pi\) radians. Therefore, the conversion factor is \(\frac{\pi}{180}\). To convert degrees to radians, you multiply the number of degrees by this factor. For example, to convert -0.46 degrees to radians, you calculate
\[ -0.46 \cdot \frac{\pi}{180} \].

Understanding this concept is crucial for solving problems involving angle measures in different units, and it is widely used in calculations involving periodic functions, like sine and cosine, which naturally operate in radians.

Radian Measure

The radian measure is a way of expressing angles based on the radius of a circle. It's an SI derived unit which makes calculations with angles straightforward, especially in calculus and physics. This is because the radian is a 'natural' unit of angular measure, derived from the properties of the circle, unlike the degree which is an arbitrary subdivision of a circle into 360 parts.

One full rotation around a circle is always 2\(\pi\) radians, regardless of the circle's size. This makes radian measures directly related to the geometry they describe, not to the degrees' artificial division. When working with radians, it's often useful to know common angle equivalents, like \(\pi\) radians equals 180 degrees, and \(\pi/2\) radians is 90 degrees. These reference points can help visualize and understand angles expressed in radians.

Significant Figures

In mathematics and science, significant figures are used to indicate the precision of a measured or calculated quantity. When converting angles from degrees to radians and rounding the results, it is imperative to pay attention to significant figures to ensure the accuracy of the answer.

For instance, when the angle -0.46 degrees is converted to radians, it requires us to retain a certain number of significant figures in the final result, often determined by the context or instructions provided. If the problem specifies that the answer should be rounded to three decimal places, we will adjust the result accordingly. This usually involves looking at the fourth decimal place and rounding up if this digit is 5 or higher, or rounding down if it is less than 5. Adhering to the rules of significant figures ensures that the solution reflects appropriate precision based on the given data.