Problem 3 - Technical Mathematics with Calculus

Question

Convert to radians. $$35.25^{\circ}$$

Step-by-Step Solution

Answer

\(\frac{35.25}{180} \pi\) radians

Step 1: Understand the Conversion Factor

Since there are \(2\pi\) radians in \(360^\circ\), the conversion factor from degrees to radians is \(\frac{\pi}{180}\). This factor is used to convert degrees to radians by multiplying the angle in degrees by this factor.

Step 2: Apply the Conversion Factor

Multiply the given angle in degrees by the conversion factor to find the angle in radians.\[35.25^\circ \times \frac{\pi}{180}\]

Step 3: Perform the Multiplication

Perform the multiplication to get the angle in radians.\[\begin{align*}35.25^\circ \times \frac{\pi}{180} &= \frac{35.25\pi}{180} \&= \frac{35.25}{180} \pi\end{align*}\]

Key Concepts

Trigonometry

Trigonometry is a branch of mathematics that deals primarily with triangles, particularly right-angled triangles. It is concerned with the study of the relationships between the angles and sides of triangles. The functions sine, cosine, and tangent are foundational concepts in trigonometry and are used to describe the ratios of sides in a right-angled triangle in relation to its angles.

The application of trigonometry extends beyond triangles, influencing various scientific fields such as physics, engineering, and astronomy. One crucial application is in the measurement and conversion of angles, a concept we encounter when we need to switch between degrees and radians.

Radian Measure

In radian measure, angles are not measured in degrees but in radians. A radian is the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle. There are exactly \(2\pi\) radians in a complete turn, which equals 360 degrees.

Using radian measure can simplify many mathematical expressions, particularly in calculus, and it's the standard unit for angle measurement in most branches of mathematics. It's particularly important for students to grasp this concept as it often leads to a more intuitive understanding of angle-related functions when applied to real-world scenarios.

Angle Conversion

Converting an angle from degrees to radians involves using a conversion factor. This factor, \(\frac{\pi}{180}\), is derived from the relationship that \(360^\circ\) is equivalent to \(2\pi\) radians. To convert degrees to radians, you multiply the number of degrees by \(\frac{\pi}{180}\), as seen in the example problem where we convert 35.25 degrees.

To fully understand angle conversion, it's beneficial to recall that this conversion factor simplifies comparisons and calculations that involve trigonometric functions. Always remember that one degree is equal to \(\frac{\pi}{180}\) radians and conversely, one radian is equal to \(\frac{180}{\pi}\) degrees. These relationships are vital for converting angles and applying trigonometric concepts across different applications.