Problem 18 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The angle measures \(-270°\).

Step 1: Understand Radian and Degree Relationship

Radian and degree are two units of measuring angles. The conversion from radians to degrees is based on the relationship that \( 2\pi \) radians is equivalent to \( 360° \). Therefore, \( \pi \) radians is equivalent to \( 180° \).

Step 2: Set Up the Conversion Equation

To convert a radian measure to degrees, you multiply the radian measure by \( \frac{180}{\pi} \). For \( -\frac{3\pi}{2} \) radians, the conversion equation is: \[ -\frac{3\pi}{2} \times \frac{180}{\pi} \]

Step 3: Simplify the Conversion Equation

Simplify the expression by canceling out \( \pi \) from the numerator and denominator. Then multiply \( -\frac{3}{2} \) by \( 180 \). This results in: \[ -\frac{3 \times 180}{2} = -\frac{540}{2} \]

Step 4: Compute the Final Degree Measure

Calculate the division \( -\frac{540}{2} = -270 \). Thus, the angle \( -\frac{3\pi}{2} \) radians is equivalent to \(-270°\).

Key Concepts

Angle Measurement

Angle measurement plays a vital role in both mathematics and everyday life, from simple tasks like measuring a piece of paper to complex problems in engineering and astronomy. Angles are typically measured in degrees or radians.

Degrees are perhaps the most familiar unit, often seen in a circular context, where a full circle is 360 degrees. Conversely, radians might be less familiar but are incredibly important in the world of higher mathematics. The radian measure is deeply connected to the properties of a circle.

Understanding these measurements and their conversions is crucial for solving problems that involve rotational and circular motion. Whether referencing a clock, a pizza slice, or a playground merry-go-round, the concepts of angle measurement remain the backbone of interpreting these circular phenomena.

Radian Measure

Radian measure is a way of defining angles based on the radius of a circle. When an angle's arc is equal to the circle's radius, the angle measures one radian. This creates a direct relationship between the angle and the arc length, making radians very useful in calculus and trigonometry.

Here's why radians are so effective:

• They relate directly to linear measurements (such as arc length).
• Simplify calculations in trigonometric functions.
• Offer consistent and direct integration results across mathematics and physics.

To grasp radians fully, visualize splitting a circle into sections, where you see each section as a fraction of the radius. One complete circle is equivalent to \(2\pi\) radians, making conversion between radians and degrees essential for complex problem-solving across various fields.

Degree Measure

Degree measure is another way to quantify angles, primarily based on dividing a circle into 360 equal parts. This division into degrees makes sense historically and practically, as it's easy to understand and apply. Many commonplace tools like protractors use degrees because they offer precise and accessible ways to measure angles.

Converting between radians and degrees is a method you'll frequently encounter and need to master in mathematics.
To convert radians to degrees, use the relationship that \(\pi\) radians equals \(180°\).

• Multiply the radian value by \(\frac{180}{\pi}\) to get degrees.
• This conversion is pivotal in applying trigonometric functions in degrees.
• It also helps translate real-world contexts into mathematical problems for solution finding.

Understanding degree measure and being able to switch between it and radians broadens your capability to tackle both practical and theoretical mathematical challenges.