Problem 20 - Precalculus Mathematics for Calculus

Question

Find the degree measure of the angle with the given radian measure. $$-2$$

Step-by-Step Solution

Answer

The angle is approximately \(-114.59\) degrees.

Step 1: Understanding the Conversion Formula

To convert an angle from radians to degrees, use the conversion formula: degrees = radians \( \times \frac{180}{\pi} \). This involves multiplying the radian measure by \( \frac{180}{\pi} \).

Step 2: Substituting Values

Substitute the given radian measure into the conversion formula. Here, the radian measure is \(-2\), so calculate \(-2 \times \frac{180}{\pi}\).

Step 3: Doing the Calculation

Perform the multiplication to convert the radians to degrees. Calculate \(-2 \times \frac{180}{3.14159}\) which approximately equals \(-114.59\).

Step 4: Rounding

Typically, angles in degrees are rounded to one or two decimal places. Thus, round \(-114.59\) to \(-114.59\) degrees.

Key Concepts

angle measurement

Angles are fundamental in geometry and trigonometry. They are a measure of rotation and can be expressed in two primary units: degrees and radians.
Understanding these units is key when working with angles in mathematics and its applications, such as physics and engineering.

• Radians: A radian is a unit of angle measurement that relates an angle to the radius of a circle. It is defined such that a full circle has an angle of \(2\pi\) radians.
• Degrees: A degree splits a full circle into 360 equal parts, so a circle contains 360°.

By grasping how radians and degrees represent angles, you can seamlessly switch between these units during calculations.
They enable you to interpret angles with precision in various mathematical contexts.

degree measure

The degree measure is a popular way to describe angles in many contexts, especially in everyday applications. Degrees are used because they simplify the division of a circle The reasons for adopting degrees are:

• Easy Division: Degrees divide a circle into 360 parts, which is handy for dividing tasks and visualizing angles.
• Accessibility: Degrees are easier to understand for most people, making them useful in everyday tasks such as navigation and map reading.
• Standardization: Many fields rely on degrees, providing a consistent language for discussing angles.

Despite their practicality, degrees are usually less convenient for mathematical calculations than radians, making radians preferred for calculus and trigonometry.

conversion formula

Understanding the radians to degrees conversion formula is essential when you need to interchange measurements between these two units. The formula is:\[\text{degrees} = \text{radians} \times \left(\frac{180}{\pi}\right)\]This formula allows you to calculate the equivalent degree measure from a given radian measure.
In the example provided, you have a radian measure of

• First, substitute \(-2\) into the formula: \[-2 \times \left(\frac{180}{\pi}\right)\]
• Then perform the calculation: \[-2 \times \frac{180}{3.14159} \approx -114.59\]
• Lastly, you round the result to one or two decimal places to get \-114.59°. This is how you convert radians to degrees, using this straightforward and essential formula.