Problem 70 - Algebra and Trigonometry Real Mathematics, Real People

Question

Rewrite each angle in degree measure. (Do not use a calculator.) (a) \(-\frac{15 \pi}{6}\) (b) \(\frac{28 \pi}{15}\)

Step-by-Step Solution

Answer

(a) -450 degrees, (b) 336 degrees

Step 1: Convert Radian Measure to Degree Measure

To convert from radians to degrees, use the conversion factor \(\frac{180}{\pi}\). The angle in degrees is computed by multiplying the radian measure by the conversion factor. \[ \text{angle in degrees} = \text{radian measure} \times \frac{180}{\pi} \]

Step 2: Conversion of Angle (a)

Substitute \(-\frac{15 \pi}{6}\) for the radian measure in the conversion formula to compute the angle in degrees. \[-\frac{15 \pi}{6} \times \frac{180}{\pi} = -\frac{15 \times 180}{6} = -450 \] Thus, \( -\frac{15 \pi}{6} \) radians is equivalent to -450 degrees.

Step 3: Conversion of Angle (b)

Substitute \(\frac{28 \pi}{15}\) for the radian measure in the conversion formula to compute the angle in degrees. \[\frac{28 \pi}{15} \times \frac{180}{\pi} = \frac{28 \times 180}{15} = 336 \] Thus, \( \frac{28 \pi}{15} \) radians is equivalent to 336 degrees.

Key Concepts

Radian to Degree Conversion

Converting angles from radians to degrees is a fundamental skill in trigonometry and mathematics education. The basic principle is using the conversion factor \(\frac{180}{\pi}\), which helps in transforming radian measure to degree measure. This conversion factor arises because 180 degrees is equivalent to \(\pi\) radians, establishing a direct proportion. Suppose you have an angle measured in radians; you multiply it by \(\frac{180}{\pi}\) to derive the angle in degrees.

• Formula: \[ \text{angle in degrees} = \text{radian measure} \times \frac{180}{\pi} \]
• By substituting the radian value into this formula, you convert the angle to degrees.

For example, consider the angle \(-\frac{15 \pi}{6}\). To find its degree measure, substitute it into the formula and simplify:
\(-\frac{15 \pi}{6} \times \frac{180}{\pi} = -450\). The radians simplify to -450 degrees.
This method ensures accurate and efficient conversion from radians to degrees, essential knowledge in both academic and practical applications of trigonometry.

Trigonometry

Trigonometry is a branch of mathematics that explores the relationships between the angles and sides of triangles. It is not only limited to geometry but extends into various applications in physics, engineering, and computer science. Understanding angles, whether in radians or degrees, is crucial because they describe the inclination of various lines or planes.

• Basic functions like sine, cosine, and tangent rely on angle measures to describe patterns in waves and oscillations.
• Knowledge of radians and degrees helps in calculating these functions accurately for different angles.

Angles in trigonometry can be measured in degrees or radians. Degrees are more common in low-level mathematics education, while radians are often used in higher mathematics because they simplify many formulas. Recognizing and converting between these units is vital for solving trigonometric problems and understanding the function graphs.

Mathematics Education

In mathematics education, understanding the conversion and relationships between different units of measure is crucial. Learning about angle measures in radians and degrees helps students build a solid foundation in various mathematical domains, including calculus and beyond.

• Teaching students how to convert radians to degrees (and vice versa) develops their problem-solving skills.
• Practical problems, like those involving trigonometric equations, often require the flexibility of switching between units.

In educational settings, explaining these concepts with various examples and practice problems helps ensure that students grasp the conversion process. Visual aids, such as the unit circle, demonstrate the relationship between radians and degrees effectively, making the abstract nature of these conversions more tangible. Consistent practice and real-world applications further aid students in understanding, ensuring they can apply these skills in diverse scenarios.