Problem 3 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

Convert each of the following radian measurements to degrees: a. \(0.25\) radian b. 1 radian c. \(-1.5\) radians

Step-by-Step Solution

Answer

0.25 radians ≈ 14.324 degrees, 1 radian ≈ 57.2958 degrees, -1.5 radians ≈ -85.9437 degrees.

Step 1: Understand the Conversion Formula

To convert radians to degrees, use the formula: \[ \text{degrees} = \text{radians} \times \frac{180}{\text{π}} \] This formula uses the fact that 1 radian is equal to \( \frac{180}{\text{π}} \) degrees.

Step 2: Convert 0.25 Radians to Degrees

Substitute 0.25 into the formula: \[ 0.25 \times \frac{180}{\text{π}} = 0.25 \times 57.2958 \text{ degrees} \ = 14.324 \text{ degrees} \]

Step 3: Convert 1 Radian to Degrees

Substitute 1 into the formula: \[ 1 \times \frac{180}{\text{π}} = 1 \times 57.2958 \text{ degrees} \ = 57.2958 \text{ degrees} \]

Step 4: Convert -1.5 Radians to Degrees

Substitute -1.5 into the formula: \[ -1.5 \times \frac{180}{\text{π}} = -1.5 \times 57.2958 \text{ degrees} \ = -85.9437 \text{ degrees} \]

Key Concepts

angle conversion

In trigonometry, it's often necessary to switch between different units of angles: radians and degrees. This process is known as angle conversion. Understanding how to convert angles between these units can be very useful for solving trigonometric problems and for clarity in communication across different contexts.

trigonometry

Trigonometry is a branch of mathematics that studies the relationships between the angles and sides of triangles. It forms the foundation for understanding many principles in various fields like physics, engineering, and even art. Basic trigonometric functions such as sine, cosine, and tangent are fundamental in many calculations and are frequently used in angle conversion tasks.

radian measurement

A radian is a measure of angle used in mathematics. It is one of the two standard units for measuring angles, the other being degrees. One radian is defined as the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle. The whole circle is equal to 2π radians. This makes radians especially useful in calculus and other advanced math topics because they simplify many formulas.

degree measurement

Degrees are another common unit of angle measurement. A full circle is divided into 360 degrees, making it a bit more familiar in everyday contexts such as navigation and geography. To change from degrees to radians or vice versa, you can use simple conversion formulas. The key is to remember that 180 degrees equal π radians.