Problem 117 - Algebra and Trigonometry Real Mathematics, Real People

Question

Convert the angle measure from degrees to radians. Round your answer to three decimal places. $$-215.63^{\circ}$$

Step-by-Step Solution

Answer

The measure in radians is approximately -3.763 radians.

Step 1: Identify the given degree measure

The given degree measure is -215.63 degrees.

Step 2: Use the conversion formula

The conversion formula to convert from degrees to radians is \( radianMeasure = degreeMeasure * \frac{\pi }{180} \). Therefore, you need to apply this formula by substituting -215.63 in place of the \( degreeMeasure \).

Step 3: Calculate and round off

Now, perform the multiplication and divide by 180, then round off the answer to three decimal places.

Key Concepts

Understanding Trigonometry

Trigonometry is a branch of mathematics that focuses on the relationships between angles and sides of triangles. It's vital in fields like physics, engineering, astronomy, and even in everyday tasks such as navigation and construction.

At its core, trigonometry deals with the study of sine, cosine, tangent, and other trigonometric functions that represent the relationship between the angles and lengths of a triangle's sides. These functions have various applications, such as calculating distances that are not easily measured and understanding the motion of waves.

Navigating Angle Conversion

Angle conversion is a fundamental aspect of trigonometry where angles measured in one unit, commonly degrees, are converted into another unit, such as radians. Since different fields employ different units of angular measurement, the ability to convert between these units is essential.For instance, while degrees are widely used in navigation and drafting, radians are the preferred unit in calculus and physics due to their natural fit with certain mathematical expressions. Radians measure angles based on the radius of a circle, making them a more 'universal' unit that directly relates the angle to the arc length in a circle.

Radian Measure Explained

The radian measure is a way of expressing angles that are derived from the circumference of a circle. One radian is the angle created when the arc length is equal to the radius of the circle. Since the total circumference of a circle is equal to \( 2\pi \) times the radius, a circle is said to have \( 2\pi \) radians.To convert degrees to radians, we use the fact that \( 360^\circ \) is equal to \( 2\pi \) radians. Hence, \( 1^\circ = \frac{\pi}{180} \) radians. Therefore, by multiplying a degree measure by \( \frac{\pi}{180} \) you can convert it to radians. This conversion is crucial in the application of trigonometry, as many trigonometric functions require inputs to be in radians for calculations, especially in integration and differentiation within calculus.