Problem 21 - Precalculus Mathematics for Calculus

Question

Find the degree measure of the angle with the given radian measure. $$-1.2$$

Step-by-Step Solution

Answer

The angle is approximately \(-68.75\) degrees.

Step 1: Understand the Radian Measure

The given radian measure of the angle is \(-1.2\). Recall that radian measure is based on the radius of a circle, where \(2\pi\) radians is one full revolution (360 degrees). We need to convert this radian measure into degrees.

Step 2: Use the Conversion Formula

The conversion formula to change radians into degrees is \(1 \, ext{radian} = \frac{180}{\pi} \, ext{degrees}\). We will use this formula to convert \(-1.2\) radians to degrees.

Step 3: Perform the Calculation

Multiply \(-1.2\) by the conversion factor \(\frac{180}{\pi}\): \[-1.2 \times \frac{180}{\pi} = -1.2 \times 57.2958 \approx -68.75496\] Round this to two decimal places if necessary, making it approximately \(-68.75\) degrees.

Step 4: Interpret the Result

The resulting negative degree measure means that the direction of measurement is clockwise from the positive x-axis. Hence, the angle \(-1.2\) radians is equivalent to approximately \(-68.75\) degrees.

Key Concepts

Radian Measure

Radian measure is central to circle geometry and trigonometry. But what does a radian actually represent? Simply put, a radian measures the angle created when the radius is wrapped around the edge of the circle. Imagine taking the length of the radius and bending it along the circumference of the circle. The angle subtender is 1 radian.

This angle measure is tied directly to the circle's properties. A full circle is equal to 2π radians because the circumference of a circle is 2π times its radius. Here are a few key points about radians to remember:

• Radians are dimensionless and are natural in mathematics.
• One full circle revolution is 2π radians.
• They are used extensively in calculus, physics, and engineering.

Understanding the concept of radian measure makes it easier to transition into angle conversion.

Degree Measure

When discussing angles outside of purely mathematical contexts, we often refer to degree measure. The degree is another way to quantify angles, where a full circle is divided into 360 equal parts, known as degrees.

Each degree can also be subdivided into minutes and seconds, though in many practical applications, we'll use decimal fractions. Degrees are typically used for:

• Navigational purposes such as compass directions.
• Simple geometry problems.
• Everyday references to angles.

Degrees are particularly user-friendly for those who do not work directly in fields like engineering or physics. However, to switch between radians and degrees effectively, mastering both systems is invaluable.

Circle Geometry

Circle geometry is a fascinating area of mathematics that deals with the properties and measures of circles and their components. Here, understanding the relationship between angles and the circle's circumference becomes key.

Within circle geometry, angles can be expressed in either degrees or radians. Why is this important? Consider these fundamental aspects:

• The circumference of a circle is 2π times the radius.
• Radian measure seamlessly relates the radius to the circular arc.
• Degrees and radians offer unique perspectives depending on the problem context.

Circle geometry applications extend from basic geometry tasks to complex equations in physics. Mastery of this topic enhances understanding of conversion concepts.

Angle Conversion

Converting between different angle measures is a key skill when working with circles. Angle conversion typically involves changing radians to degrees or vice versa. Understanding this conversion improves your ability to interpret angles in different contexts. Let's break down the process.

The crucial relation for conversion is that π radians equal 180 degrees. This leads us to the conversion formula:

• 1 radian = \( \frac{180}{\pi} \) degrees
• 1 degree = \( \frac{\pi}{180} \) radians

By applying these formulas, you can easily change between radians and degrees. For example, to convert -1.2 radians to degrees, you multiply -1.2 by \( \frac{180}{\pi} \), resulting in approximately -68.75 degrees. Remember, a negative result indicates clockwise rotation. Mastering angle conversion broadens your comprehension of geometric problems.