Problem 39 - Precalculus with Calculus Previews

Question

In Problems \(33-40,\) convert the given angle from radians to degrees. $$ 3.1 $$

Step-by-Step Solution

Answer

3.1 radians is approximately 177.595 degrees.

Step 1: Understanding the Conversion Factor

To convert an angle from radians to degrees, we use the conversion factor: \[1 ext{ radian} = \frac{180}{ ext{π}} ext{ degrees} \] This means that multiplying the radian measure by \( \frac{180}{ ext{π}} \) will give us the angle in degrees.

Step 2: Apply the Conversion

Apply the conversion factor to the given angle:\[3.1 ext{ radians} \times \frac{180}{ ext{π}} ext{ degrees/radian} \]

Step 3: Perform the Multiplication

Perform the multiplication by splitting it into steps:1. Multiply 3.1 by 180: \[3.1 \times 180 = 558 \]2. Divide the result by π: \[\frac{558}{ ext{π}}\] Use the approximate value for π (3.14159) to get a numerical result when calculated on a calculator, \[\frac{558}{3.14159} \approx 177.595 \]

Step 4: Final Result

Therefore, the angle in degrees is approximately 177.595 degrees.

Key Concepts

Radians to Degrees

Converting angles from radians to degrees is a common task in trigonometry, and it's crucial for understanding various mathematical applications. When an angle is given in radians, it needs to be converted to degrees to make it more intuitive for everyday use. This conversion is not just a matter of convenience; it's also essential for consistency across measurements. Remember that a full circle constitutes both 360 degrees and \(2\pi\) radians, which provides the basis for the conversion between these two units.

Conversion Factor

The conversion factor between radians and degrees is derived from the relationship between a full circle's rotation measures. Specifically, you need to remember:

• 1 radian \(= \frac{180}{\pi}\) degrees
• This means that to convert radians to degrees, you multiply the radian value by \(\frac{180}{\pi}\)

This conversion factor is universally applicable, allowing you to convert any angle from radians to degrees with ease. This approach ensures that mathematical calculations and measurements can be standardized across different fields and applications.

Angle Measurement

When dealing with angles, it's important to have a solid understanding of measurement units. Angles measure the amount of turn or rotation and can be expressed in different units, primarily degrees and radians. Degrees are more commonly used in daily life, like in navigation or in describing weather patterns. Radians, however, are preferred in mathematical calculations, particularly in calculus and physics, because they simplify many equations and formulas due to their relation to the arc length of a circle.

Trigonometry

Trigonometry is the mathematics of triangles, and it extensively uses radian and degree measures. It deals with relationships between the angles and sides of triangles, especially right triangles. Knowledge of converting radians to degrees is crucial here since many trigonometric functions, such as sine, cosine, and tangent, are often calculated using radians in scientific settings. Understanding trigonometry is not only essential for solving geometric problems but also for fields like engineering, physics, and even computer graphics, where angle measures greatly influence computations and visuals. The seamless conversion between radians and degrees aids in these diverse applications, making angle measurement more versatile and widely applicable.