Problem 53 - Precalculus : Building Concepts and Connections

Question

Convert each angle from degrees to radians. $$-150^{\circ}$$

Step-by-Step Solution

Answer

The angle -150 degrees is equivalent to \(-\frac{5\pi}{6}\) radians in terms of radians.

Step 1: Understand the conversion relationship

Firstly, understand that 180 degrees is equivalent to \(\pi\) radians. This ratio can be used to convert degrees to radians.

Step 2: Setup the conversion equation

To convert degrees to radians, you can multiply the degrees by the ratio of \(\pi\) radians over 180 degrees. So, the conversion equation should look like this: \(-150^{\circ} * \(\frac{\pi}{180}\)\)

Step 3: Simplify the conversion equation

Calculating, -150 *\(\frac{\pi}{180}\) simplifies to \(-\frac{5\pi}{6}\) radians

Key Concepts

Angle Conversion

Understanding angle conversion is crucial when studying geometry and trigonometry. In essence, angle conversion is the process of changing the measurement of an angle from one unit to another. The most common units of angle measurement are degrees and radians. Learning to convert between these units is a fundamental skill in mathematics.

Radian Measure

A radian is a unit of angular measure used in mathematics. It is the standard unit of angular measure, used in many areas of mathematics. One radian is the angle created when the radius of a circle is wrapped along the circumference of the circle. An important aspect to grasp is that the circumference of a circle is always \(2\pi\) times the radius, which directly gives us the relationship that there are \(2\pi\) radians in one complete revolution of a circle, or 360 degrees. Thus, understanding radian measure is not only about knowing its definition but also about seeing its relation to the geometry of circles.

Pi Radian Relationship

The relationship between Pi (\(\pi\)) and radians is vital for converting degrees to radians. Pi (\(\pi\)) is a mathematical constant that represents the ratio of a circle's circumference to its diameter, and it is approximately 3.14159. Because the circumference of a circle is \(2\pi\) times the radius, and a full rotation around a circle is 360 degrees, we have the important relationship that 180 degrees is equivalent to \(\pi\) radians. This relationship forms the backbone of angle conversion, as it allows us to create a conversion factor by setting degrees over radians (\(\frac{180^\circ}{\pi}\)) and using this to convert angles from degrees to radians and vice versa.

Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between the angles and sides of triangles. It has a vast number of applications in various fields such as physics, engineering, astronomy, and even art. The trigonometric functions, like sine, cosine, and tangent, are defined based on the angles of a right triangle or the unit circle, and they are crucial tools in calculating distances, angles, and helping to model periodic phenomena. Being fluent in converting degrees to radians is paramount in trigonometry since many trigonometric functions use radian measure as their argument.

When you're working through trigonometric problems, always remember to check if your calculator is set to the correct unit (degrees or radians) to ensure accurate results. Radians are often the preferred unit in higher mathematics because they simplify many formulas and concepts, providing a more natural way to express angles and periodicity.