Problem 16 - Algebra and Trigonometry

Question

\(13-24\) . Find the degree measure of the angle with the given radian measure. $$ -\frac{3 \pi}{2} $$

Step-by-Step Solution

Answer

-270 degrees

Step 1: Understand the Problem

We need to convert the given radian measure \(-\frac{3 \pi}{2}\) to degrees.

Step 2: Recall the Conversion Formula

The formula to convert radians into degrees is \( \, Degrees = Radians \times \frac{180}{\pi} \, \).

Step 3: Substitute the Values Into the Formula

Substitute \(-\frac{3 \pi}{2}\) into the formula: \[ Degrees = \left(-\frac{3 \pi}{2}\right) \times \frac{180}{\pi} \]

Step 4: Simplify the Expression

Calculate the expression: \[ = -\frac{3 \times 180}{2} \]

Step 5: Perform the Multiplication and Division

Simplify further by performing the multiplication and division: \[ = -\frac{540}{2} = -270 \]

Step 6: Conclude the Conversion

The angle of \(-\frac{3 \pi}{2}\) radians is equivalent to \(-270\) degrees.

Key Concepts

Radians

Radians are a way to measure angles used especially in mathematics and physics. Unlike degrees, which divide a circle into 360 parts, radians relate angles to the radius of a circle. To better understand, picture this: if you take the radius of a circle and wrap it along the edge of the circle, the angle you get in radians is precisely the distance the radius covers on the circle's arc.
Radians provide a natural way to handle circles and trigonometric functions because the length of an arc directly equals the angle in radians times the radius. This simplicity makes it incredibly useful for things like calculus.

• One full circle equals 2π radians.
• Half a circle, or a straight angle, equals π radians.
• A quarter of a circle, or a right angle, equals π/2 radians.

Understanding this basis helps when converting between radians and degrees, ensuring accurate calculations and interpretations of angle sizes.

Degrees

Degrees are another unit of angle measurement and are more commonly used in everyday life. The degree system divides a circle into 360 equal parts. This division comes from ancient civilizations, which used base 60 numeration. Using degrees is straightforward for most everyday applications, such as geometry, navigation, and geography.
It's beneficial because:

• An entire circle is 360 degrees.
• A straight line represents 180 degrees.
• A right angle is 90 degrees.

Although degrees aren't always as mathematically elegant as radians, they're practical for various fields and help provide a simpler perspective on angle measurement.

Angle Measurement

Angle measurement is crucial for understanding the size and orientation of angles in mathematics. An angle is defined as the amount of turn between two arms converging at a vertex. Both radians and degrees measure how steeply these two arms deviate from each other.
- When you measure angles in mathematics, physics, or engineering, radians are often preferred because they make simple relationships in calculus. - In contrast, degrees are generally more intuitive for practical measurements like those in design, architecture, and standard geometry problems.
Both methods essentially depict the same notion—the extent of an angle's curve or incline—but differ in how they are scaled. Knowing the difference and application of degrees and radians can significantly improve computational skills.

Conversion Formula

To transition between radians and degrees, you need to understand and apply the conversion formula. The conversion formula is essential because it bridges the two measurement systems, enabling the expression of angles in terms that suit specific applications more efficiently.
The formula reads as follows:- To convert from radians to degrees, multiply the radian measure by \( \frac{180}{\pi} \).For example, the given radian measure \(-\frac{3 \pi}{2}\):

• Multiply by \( \frac{180}{\pi} \)
• This results in \(-270\) degrees

This formula facilitates operations in different fields, ensuring a smooth transition between systems and fostering a deeper understanding of angles in diverse contexts. Recognizing when and how to use this conversion aids students in solving a variety of problems seamlessly.