Problem 33 - Algebra and Trigonometry

Question

In Exercises \(29-34,\) convert each angle in degrees to radians. Round to two decimal places. $$ 200^{\circ} $$

Step-by-Step Solution

Answer

The angle 200 degrees is approximately \(3.49\) radians when rounded to two decimal places.

Step 1: Recognizing the conversion factor

The first thing to note is that we have a standard conversion factor from degrees to radians, which is \(\frac{\pi}{180}\) radians for every degree.

Step 2: Applying the conversion factor

To convert the angle from degrees to radians, we multiply the given angle by the conversion factor. The calculation is: \(200 * \frac{\pi}{180}\)

Step 3: Calculating the result

Performing the multiplication operation, we find that the angle in radians is \(3.49\) (rounded to 2 decimal places).

Key Concepts

Trigonometry

At the heart of trigonometry lies the study of relationships involving lengths and angles of triangles. Essential to many fields, including physics, engineering, and even art, trigonometry paves the way for understanding waves, oscillations, and circles. A core element is the concept of angle measurements—how large or small an angle is—which leads us to compare and convert angles using different units, such as degrees and radians.

Radian Measure

Radian measure is a way of expressing angles that is based on the radius of a circle. One radian is the angle created when the arc length equal to the radius of the circle subtends that angle at the center. Unlike degrees, which are based on dividing a circle into 360 parts, radians give a direct connection between the angle and the arc length, which can be incredibly useful in advanced mathematics and calculus. For instance, when you wrap the radius of a circle around its circumference, it will fit approximately 6.28318 times, which is equivalent to \(2\pi\) radians.

Conversion Factor

A conversion factor is a numerical multiplier that allows us to change units from one system to another. In this context, to convert angle measurements from degrees to radians, we use the conversion factor \(\frac{\pi}{180}\) radians per degree. This factor is derived from the full circle relationship where \(360^\circ\) is equivalent to \(2\pi\) radians. Understanding and applying the correct conversion factor is crucial in making accurate transitions between units.

Angle Measurements

Angle measurements are used to describe the size of an angle. In schools, we commonly learn about angles in degrees, but in higher mathematics, angles in radians are often more useful. Angles measured in degrees indicate what fraction of a full circle's rotation the angle represents with a full circle being \(360^\circ\). Radians, however, relate an angle to the radius of the circle, which makes them indispensable in trigonometry and calculus, especially when dealing with periodic functions like sine and cosine.