Problem 17 - Technical Mathematics with Calculus

Question

Convert to degrees (decimal). $$1.14 \mathrm{rad}$$

Step-by-Step Solution

Answer

Approximately 65.28 degrees

Step 1: Understand the relationship between radians and degrees

To convert radians to degrees, use the fact that a full circle is 360 degrees, which is equivalent to \(2\pi\) radians. Hence, the conversion factor from radians to degrees is \(\frac{180}{\pi}\).

Step 2: Set up the conversion formula

To convert the given value in radians to degrees, multiply by the conversion factor \(\frac{180}{\pi}\). The formula is: \[\text{degrees} = \text{radians} \times \frac{180}{\pi}\]

Step 3: Calculate the value in degrees

Substitute the given value into the conversion formula: \[1.14 \times \frac{180}{\pi} = \frac{1.14 \times 180}{\pi} \approx 65.28^\circ\] Thus, \(1.14\) radians is approximately \(65.28^\circ\).

Key Concepts

Understanding Radians

Radians are a unit of angular measure used in mathematics, particularly in trigonometry. Unlike degrees, which are based on dividing a circle into 360 equal parts, radians are based on the radius of a circle. One radian is defined as the angle made at the center of a circle by an arc whose length is equal to the radius of the circle. An important aspect to grasp is that the circumference of a circle is \(2\pi\) times its radius. Therefore, a full circle, which is \(360^\circ\), corresponds to \(2\pi\) radians. To visualize this, imagine wrapping the radius of a circle around its circumference. It would take slightly more than six radii to do so, as \(2\pi\) is approximately 6.28.

The Concept of Degrees

Degrees are a more commonly known unit for measuring angles, originating from ancient civilizations dividing the circle into 360 parts for simplicity. Each degree is divided into 60 minutes, and each minute is divided into 60 seconds, which allows for precise measurements. Degrees are denoted by the symbol \(^\circ\). This unit is widely used in various fields such as geography, astronomy, and in everyday language when referring to temperatures or angles. To connect this with everyday experiences, when we say that two streets intersect at a \(90^\circ\) angle, we mean they form a right angle, which is traditionally one-quarter of a full circle.

Unit Conversion from Radians to Degrees

Unit conversion between radians and degrees allows mathematicians and scientists to toggle between the two systems. A general formula for converting radians to degrees is to multiply the number of radians by \(\frac{180}{\pi}\). On the other hand, to convert degrees to radians, you would multiply the number of degrees by \(\frac{\pi}{180}\). When performing these conversions, it's pivotal to understand that you're not changing the value of the angle, but rather expressing the same angle in a different unit.

For instance, a right angle is \(\frac{\pi}{2}\) radians or \(90^\circ\); both are the same angle but expressed in different units. It's also worth noting for students that when doing calculations or using trigonometric functions in a calculator, knowing whether the calculator is set to radians or degrees is crucial for getting the correct answer.

Trigonometry Basics

Trigonometry is a branch of mathematics dealing with the relationships between the angles and sides of triangles, particularly right-angled triangles. The key functions in trigonometry are sine, cosine, and tangent, which are ratios of the sides of a right-angled triangle compared to one of its non-right angles. These functions are fundamental in various applications ranging from physics to engineering and beyond.

Trigonometric values can be expressed and calculated in both degrees and radians. However, different applications may require different units. Radians are often preferred in calculus and many areas of science and engineering because they result in simpler and more elegant formulas. For example, in calculus, the derivatives of trigonometric functions take on a particularly simple form when the functions' arguments are in radians.