Problem 16 - Algebra and Trigonometry

Question

Find the degree measure of the angle with the given radian measure. $$ \frac{11 \pi}{3} $$

Step-by-Step Solution

Answer

The angle is 660 degrees.

Step 1: Understand the Conversions

To convert an angle from radians to degrees, we use the conversion factor: \(180^\circ = \pi \text{ radians}\). This gives us the conversion formula: \(1 \text{ radian} = \frac{180^\circ}{\pi}\).

Step 2: Apply the Conversion Formula

Use the conversion formula to convert \(\frac{11\pi}{3}\) radians to degrees. Substitute into the formula: \(\frac{11\pi}{3} \times \frac{180^\circ}{\pi} = \frac{11 \times 180^\circ}{3}\).

Step 3: Simplify the Expression

Simplify the expression \(\frac{11 \times 180}{3}\). First, multiply 11 by 180 to get 1980. Then, divide 1980 by 3 to get 660.

Key Concepts

Angle Measurement

Understanding angle measurement is crucial to geometry and trigonometry. Angles can be described as the space between two intersecting lines or rays. They are a fundamental concept in mathematics, appearing in various fields and applications. There are different ways to measure angles, with the two most common being degrees and radians. - **Degrees**: This is the most familiar unit of angle measurement. A full circle is divided into 360 degrees. This concept stems from ancient Babylonian mathematics. - **Radians**: A radian is based on the radius of a circle. It is a less intuitive concept but fundamental in higher-level math, particularly in calculus and trigonometry. Each method of angle measurement is useful depending on the situation, which is why conversion between them is often necessary.

Degree Measure

Degrees are likely the angle measurement you encountered first. They are part of everyday language when discussing angles and directions. The degree measure divides a complete circle into 360 equal parts. This system is incredibly user-friendly for simple geometric calculations and communication. - An important aspect of using degrees is understanding that a right angle measures 90 degrees. - A straight line forms an angle of 180 degrees. Degrees can easily express angular differences, such as turning, rotating, or directional changes, making them a preferred choice for navigation and basic geometry.

Radian Measure

Radian measure may be less intuitive than degrees but is critical for advanced mathematics. In radiance, one complete circle is equivalent to \(2\pi\) radians. This makes the measure especially vital in calculus and trigonometry for its natural mathematical properties.- **Definition**: A radian is the angle created when the arc length equals the radius of the circle.- When dealing with circles, radians simplify many formulas and calculations, particularly in periodic functions.- Because radian measure connects directly with the circle's radius, it offers elegant solutions and expressions in mathematical analysis.Understanding the conversion between radians and degrees, given that \(\pi\) radians equal 180 degrees, enables seamless transitions in problem-solving situations. This conversion is essential for applying various mathematical concepts effectively.