Problem 71 - Precalculus: A Concise Course

Question

Convert the angle measure from degrees to radians. Round to three decimal places. $$ -0.83^{\circ} $$

Step-by-Step Solution

Answer

-0.014 radians rounded to three decimal places.

Step 1: Identify the given angle in degrees

The angle given in the exercise is \(-0.83^{\circ}\).

Step 2: Conversion from degrees to radians

To convert from degrees to radians, multiply the degree measurement by \(\frac{\pi}{180}\). So, the conversion will look like this: \[ -0.83^{\circ} * \frac{\pi}{180}\], which computes the radian measure of the given angle.

Step 3: Calculate and Round the result

Compute the multiplication to get the angle measured in radians. Finally round the result to three decimal places. This is the desired answer.

Key Concepts

Angle Measurement

The term 'angle measurement' refers to the process of determining the size of an angle. Angles can be measured in various units, but the two most common are degrees and radians. A degree ( )( ^^) is a measure of angle equal to 2. 180180thth of a revolution, which comes from dividing the circumference of a circle into 360 equal parts. In contrast, a radian is a measure that is based on the radius of a circle; one radian is the angle created when the arc length is equal to the radius of the circle.

Radians are often used in higher-level mathematics because they simplify calculations involving trigonometric functions. In practice, being fluent in converting between degrees and radians is essential for students studying geometry, trigonometry, calculus and physics, as different problems may require different units of angle measurement.

Pi Constant

The constant nn () is intrinsically linked to the concept of radians. It represents the ratio of a circle's circumference to its diameter and is approximately equal to 3.14159. Although various approximations of nn are often used for practical calculations, it's important to note that nn is an irrational number, meaning it has an infinite number of decimal places and no exact fraction representation.

Rounding Decimal Places

Rounding is a numerical process used to shorten a number to a desired number of decimal places without significantly changing its value. This makes numbers easier to work with and communicate, although it introduces a slight error due to approximation. In the world of mathematics and sciences, clear communication and precision are crucial, and rounding enables us to achieve a balance between the two. When rounding, if the digit in the next place is five or more, we round up the last retained digit. Otherwise, we leave it unchanged.

For the conversion of angle measurements from degrees to radians, calculating the exact value might result in a number with many decimal places. Therefore, rounding to a certain number of decimal places—commonly three for mathematical exercises—is often required. This not only provides a more concise answer but also aligns with practical applications where perfect precision is not always necessary.