Conversions: Degrees and Radians
Conversions between degrees and radians are essential for understanding angular measurement, which is vital in various fields such as mathematics, physics, and engineering. Angles are formed when two lines or rays meet, and historically, the unit of degrees has its roots in ancient Babylonian culture, while the concept of radians is a more modern development. A full circle is defined as 360 degrees, which equates to 2π radians, illustrating the connection between these two units.
To convert degrees to radians, one can use the formula that relates the two: multiply the degree measure by π/180. Radians are particularly useful in mathematical formulas and physical applications, as many relationships—such as those involving arc length and area of a sector—require angles to be expressed in radians for accuracy. For instance, the formula for arc length, expressed as \( \theta r \), and the derivative of sine functions in calculus hold true only when angles are measured in radians. Understanding these conversions and their applications is crucial for effective problem-solving in science and mathematics.
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Conversions: Degrees and Radians
When two straight lines or rays meet, they form an angle. Angles have been measured over the years for many reasons. The units of angular measurement vary, but most modern applications use either degrees or radians. For many reasons, it is advantageous and even necessary, to represent angles in radians. Additionally, radians are designated as the unit for plane angle measurements by the National Institute of Standards and Technology. Therefore, the conversion between the units of degrees and radians is an important concept to understand and to be able to do.
Planar angles can be thought of as a portion of a circle. A full circle, measured in degrees, is 360º. Radian angle measures are the angle that corresponds to the arc length that is created on the circle when represented by the circle’s radius. In other words, an arc length of x corresponds to an angle of x/r radians. For the full circle, this means that the angle would be 2r/r = 2 radians. The actual units for radians are meters/meters, as this angle measure is actually unitless. The word "radian," often abbreviated rad, is used to indicate that the value represents the corresponding angle. See Figure 1, which illustrates a radian angle.
Overview
The use of degree units for angles is thought to have originated with the Babylonians in approximately 2400 BC. While not substantiated, the number of degrees is thought to connect to the number of days in the calendar year at that time. The division of degrees into minutes (60 minutes per degree) and into seconds (60 seconds into a minute) follows the Sumerian’s numerical system, which was based on 60.
Radian units are a much more recent development which have a bit of controversy associated with who officially initiated the term. Thomas Muir and James Thompson coined the word "radian" in about 1870. However, previous work by Leonard Euler referenced the concept of radians (without the explicit use of the word "radian") in his book Elements of Algebra (1770) when he wrote ei = cos + isinwhich only holds true when the angle, , is represented as a radian measure.
To convert an angle measure from degrees to radians, the relationship 360º = 2 radians is used. In particular, the degree measurement x is multiplied by 2360º or, in reduced form, by 180º. For instance, 90º * 180º = 2 radians.
Being able to make this conversion is very important as many mathematical and physical relationships only hold true when the units of angle measurement are radians. Examples of this include arc length = r where is given in radians. Additionally, the formula for a sector’s area requires radian measures when considering the angle that the sector creates in the circle’s center, , A = ½ r2. The relationship between tangential linear and angular velocity, vT = r, requires that the angular velocity is represented in rad/s. In calculus, important derivative equations involving trigonometric functions, such as d/dx sinx = cosx holds true if x is in radians.
Bibliography
Epp, Susana S. Discrete Mathematics and Applications. Boston: Cengage, 2011.
Euler, Leonard. Elements of Algebra. Trans. John Hewlett. New York: Cambridge UP, 2009.
Kuhn, Thomas S. The Structure of Scientific Revolutions. Chicago: U of Chicago P, 2012.
Tiner, John Hudson. The World of Mathematics: From Ancient Record Keeping to the Latest Advances in Computers. Green Forest, AR: New Leaf, 2013.