Problem 17 - Basic Technical Mathematics with Calculus

Question

By means of the definition of a radian, change the given angles in radians to equal angles expressed in degrees to the nearest \(0.01^{\circ}\). $$0.265 rad$$

Step-by-Step Solution

Answer

The angle \(0.265\) radians is approximately \(15.18^{\circ}\).

Step 1: Understand the relationship between radians and degrees

A full circle is equal to \(360^{\circ}\) or \(2\pi\) radians. Therefore, \(1\) radian is equal to \(\frac{180}{\pi}\) degrees. Our goal is to convert the given radians to degrees using this relationship.

Step 2: Use the conversion formula

The formula to convert radians to degrees is \( \text{Degrees} = \text{Radians} \times \frac{180}{\pi} \). We'll apply this formula to the given angle of \(0.265\) radians.

Step 3: Calculate the degree equivalent

Substitute \(0.265\) into the conversion formula: \[\text{Degrees} = 0.265 \times \frac{180}{\pi} \] This evaluates to: \[\text{Degrees} = 0.265 \times \frac{180}{3.141592653589793}\] \[\text{Degrees} \approx 15.183 \text{ (to 3 decimal places)}\]

Step 4: Round to the nearest hundredth

To express the angle to the nearest hundredth of a degree, round \(15.183\) to \(15.18^{\circ}\).

Key Concepts

Radian to Degree Conversion

When working with angles, it's important to understand the two main units of measurement: radians and degrees. These are used interchangeably in mathematics, especially in trigonometry. A full circle makes a complete turn of 360 degrees, which is equivalent to \(2\pi\) radians. This shows the deep link between the two:

• 1 radian equals \( \frac{180}{\pi} \) degrees
• 1 degree equals \( \frac{\pi}{180} \) radians

To convert radians to degrees, you use the conversion formula:
\[ \text{Degrees} = \text{Radians} \times \frac{180}{\pi} \]
For instance, converting 0.265 radians involves multiplying by \( \frac{180}{\pi} \), which provides the degree measure. This allows angles to be understood and compared in different contexts.

Mathematical Relationships

Mathematical relationships are the backbone of converting angles from radians to degrees. Recognizing the relation between radians and degrees is crucial in applied mathematics. This relationship is based on the properties of circles and the constants associated with them:

• \(360^{\circ}\) is the angle of a full circle
• Zero radians correspond to zero degrees
• \(\pi\) is the connection bridge, where \(2\pi\) radians equal \(360^{\circ}\)

These fundamental relationships allow the straightforward conversion of angle measures. Understanding this conversion is essential not only for angle measurement but also for exploring deeper trigonometric principles.

Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between the angles and sides of triangles.
When dealing with trigonometric functions, it's common to switch between radians and degrees.
The unit circle is a key tool in trigonometry. It's a circle with a radius of 1 centered at the origin of a coordinate plane. Radii intersect with the circle, creating angles measured in radians. Angles in trigonometry often originate as radian measures because of their natural geometric interpretation. However, practical problems in trigonometry often require conversion to degrees. This makes understanding and applying the conversion formula essential for:

• Solving triangles
• Understanding wave functions
• Working with oscillations and rotations

By mastering radian and degree conversions, you gain a stronger foundation in the practical applications of trigonometry across various fields.