Problem 22 - Algebra and Trigonometry

Question

\(13-24\) . Find the degree measure of the angle with the given radian measure. $$ \frac{5 \pi}{18} $$

Step-by-Step Solution

Answer

50 degrees.

Step 1: Understand the Conversion Factor

To convert radians to degrees, we multiply the radian measure by \( \frac{180}{\pi} \). This is because \( \pi \) radians is equivalent to 180 degrees.

Step 2: Set Up the Conversion

The given radian measure is \( \frac{5\pi}{18} \). Set up the conversion by multiplying this by \( \frac{180}{\pi} \):\[\frac{5\pi}{18} \times \frac{180}{\pi}\]

Step 3: Simplify the Expression

Simplify the expression by canceling \( \pi \) in the numerator and denominator:\[\frac{5}{18} \times 180\]

Step 4: Perform the Multiplication

Perform the multiplication:\[\frac{5 \times 180}{18}\]Simplifying gives:\[\frac{900}{18}\]

Step 5: Calculate the Result

Divide 900 by 18:\[900 \div 18 = 50\]The degree measure of \( \frac{5\pi}{18} \) radians is 50 degrees.

Key Concepts

Radian to Degree Conversion

When delving into the world of trigonometry, you'll often encounter angle measurements in both radians and degrees. Radians and degrees are two different units used to measure angles, much like inches and centimeters are used to measure length. The conversion between these two is essential for a comprehensive understanding of trigonometry.

• A radian is a measure based on the radius of a circle; specifically, it's the angle subtended by an arc that's equal in length to the circle's radius. The entire circumference of the circle corresponds to an angle of 2times \(\pi\) radians.
• Degrees, on the other hand, divide the circle into 360 equal parts. Thus, the complete angle around a circle is 360 degrees.

To convert radians to degrees, we utilize the conversion factor \(\frac{180}{\pi}\), due to the fact that \(\pi\) radians equals 180 degrees. For example, if you have an angle measure of \(\frac{5\pi}{18}\) radians, you multiply it by \(\frac{180}{\pi}\) to convert it into degrees, as shown in the step-by-step solution. This conversion is crucial, as it allows us to move freely between the two systems, understanding angles in both contexts.

Measuring Angles

Measuring angles is a fundamental skill in math and geometry, essential for accurate calculations in various fields such as astronomy, architecture, and engineering. Angles can be measured in different units, the most common being degrees and radians.

• Degrees: A circle is divided into 360 degrees. This division is an ancient system based on the counting system used by the Babylonians.
• Radians: Radians offer a more natural measure that directly relates the arc length of a circle to its radius.

Using radians and degrees effectively often depends on the context. Degrees are straightforward and widely used in everyday applications. Radians are commonly used in higher mathematics, especially calculus and trigonometry, as they simplify many formulas and calculations. Understanding how to measure and convert these angles is crucial for students in advancing their mathematical knowledge. Each measure caters to specific needs and can offer unique advantages depending on the situation.

Trigonometry Basics

Trigonometry is a branch of mathematics that studies the relationships between side lengths and angles in triangles. It's fundamental for fields that require precise measurements and calculations, such as construction, navigation, and physics.

At its core, trigonometry introduces three primary functions:

• **Sine (sin)**: This function relates the angle to the ratio of the opposite side over the hypotenuse in a right-angled triangle.
• **Cosine (cos)**: Similar to sine, the cosine of an angle is the ratio of the adjacent side to the hypotenuse.
• **Tangent (tan)**: Tangent provides the ratio of the opposite side to the adjacent side.

These functions are crucial when engaging with problems that involve angles and require precise calculations. Understanding and using these trigonometric functions are essential skills, enabling you to solve complex problems and analyze various concepts in mathematics and beyond. Incorporating knowledge of angle measurements and conversions amplifies your capability in trigonometry, as it allows you to transition seamlessly between various mathematical systems.