Problem 20 - Algebra 2 and Trigonometry

Question

In \(13-22\) , find the degree measure of each angle whose radian measure is given. \(\frac{11 \pi}{6}\)

Step-by-Step Solution

Answer

The degree measure of \(\frac{11\pi}{6}\) radians is \(330^\circ\).

Step 1: Understand the Problem

We need to convert the radian measure \(\frac{11\pi}{6}\) into degrees. Remember that to convert from radians to degrees, we use the relationship that \(180^\circ = \pi\) radians.

Step 2: Set Up the Formula

To convert radians to degrees, use the formula:\[\text{Degrees} = \text{Radians} \times \frac{180^\circ}{\pi}\]Substitute the radian measure \(\frac{11\pi}{6}\) into this formula.

Step 3: Perform the Conversion

Calculate the degrees using the formula:\[\frac{11\pi}{6} \times \frac{180^\circ}{\pi} = \frac{11 \times 180^\circ}{6}\] The \(\pi\) terms cancel out, so it simplifies to a multiplication problem.

Step 4: Simplify the Calculation

Perform the multiplication and division:\[\frac{11 \times 180}{6} = \frac{1980}{6} = 330^\circ\] Divide 1980 by 6 to obtain the degree measure.

Key Concepts

Radians

Radians are a way to measure angles based on the radius of a circle. Unlike degrees, which divide a circle into 360 equal parts, radians are related to the circle's arc. One whole circle is equal to \( 2\pi \) radians. This means that if you travel around the entire circumference of a circle, you have traveled through \( 2\pi \) radians.

To understand radians, imagine wrapping the radius of the circle around its edge. This wraps around the circle in about 6.28 times because \( 2\pi \approx 6.28 \).

• 1 radian is the angle created when the arc length equals the radius.
• Radians provide a direct connection between linear and angular measurements.
• Because radians are based on \( \pi \), they simplify calculations in trigonometry and calculus.

Understanding radians helps bridge the gap between linear movement and rotational motion, making them a key concept in trigonometry.

Degrees

Degrees are perhaps the most commonly known units for measuring angles, dividing a circle into 360 equal parts. This division stems from ancient astronomy and has been retained due to its ease of use and divisibility by many numbers.

Each degree represents \(\frac{1}{360}\) of a full circle. In more practical terms:

• 90 degrees form a right angle.
• 180 degrees create a straight line, or half circle.
• 270 degrees account for three quarters of the circle's rotation.

Because degrees are so widely known and used, they are often the starting point for learning about angles and trigonometry. However, in higher-level mathematics, radians become more useful due to their natural fit with mathematical constants like \( \pi \).

Angle Measurement

Angle measurement is a fundamental concept that helps describe how 'open' an angle is. It determines the amount of rotation needed to bring one line into alignment with another. In both degrees and radians, angles describe how far you turn from your starting point.

Generally, angles can measure:

• Acute angles, which are less than 90 degrees.
• Right angles, exactly 90 degrees.
• Obtuse angles, greater than 90 degrees but less than 180 degrees.
• Straight angles, exactly 180 degrees.
• Reflex angles, greater than 180 degrees.

The understanding of angle measurement is applicable in real-world contexts like navigation, construction, and even art, as it helps transform abstract concepts into practical solutions.

Trigonometry

Trigonometry is the branch of mathematics that explores the relationships between the sides and angles of triangles. It primarily deals with right-angled triangles, but can also be applied to other shapes and situations.

Key functions in trigonometry include:

• Sine (sin), which relates the opposite side to the hypotenuse.
• Cosine (cos), which compares the adjacent side to the hypotenuse.
• Tangent (tan), which examines the opposite side over the adjacent side.

These functions are crucial for solving problems involving waves, circular objects, and oscillations, as they provide a link between angle measurements and physical distances. Thus, trigonometry is indispensable in fields such as physics, engineering, and computer graphics.