# Problems and Solutions in Plane Trigonometry (LaTeX Edition): For the use of Colleges and Schools

The concept of radian measure, as opposed to the degree of an angle, is normally credited to Roger Cotes in Prior to the term radian becoming widespread, the unit was commonly called circular measure of an angle. The idea of measuring angles by the length of the arc was already in use by other mathematicians.

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For example, al-Kashi c. The term radian first appeared in print on 5 June , in examination questions set by James Thomson brother of Lord Kelvin at Queen's College , Belfast. He had used the term as early as , while in , Thomas Muir , then of the University of St Andrews , vacillated between the terms rad , radial , and radian. In , after a consultation with James Thomson, Muir adopted radian. Longmans' School Trigonometry still called the radian circular measure when published in The International Bureau of Weights and Measures  and International Organization for Standardization  specify rad as the symbol for the radian.

So, for example, a value of 1. For example,. In calculus and most other branches of mathematics beyond practical geometry, angles are universally measured in radians.

This is because radians have a mathematical "naturalness" that leads to a more elegant formulation of a number of important results. Most notably, results in analysis involving trigonometric functions are simple and elegant when the functions' arguments are expressed in radians.

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For example, the use of radians leads to the simple limit formula. In all such cases it is found that the arguments to the functions are most naturally written in the form that corresponds, in geometrical contexts, to the radian measurement of angles.

Mathematically important relationships between the sine and cosine functions and the exponential function see, for example, Euler's formula are, again, elegant when the functions' arguments are in radians and messy otherwise. Although the radian is a unit of measure, it is a dimensionless quantity. This can be seen from the definition given earlier: the angle subtended at the centre of a circle, measured in radians, is equal to the ratio of the length of the enclosed arc to the length of the circle's radius. Since the units of measurement cancel, this ratio is dimensionless.

Although polar and spherical coordinates use radians to describe coordinates in two and three dimensions, the unit is derived from the radius coordinate, so the angle measure is still dimensionless. The radian is widely used in physics when angular measurements are required. Likewise, the phase difference of two waves can also be measured in radians. Metric prefixes have limited use with radians, and none in mathematics. The divergence of laser beams is also usually measured in milliradians.

An approximation of the trigonometric milliradian 0. For the small angles typically found in targeting work, the convenience of using the number in calculation outweighs the small mathematical errors it introduces. Similarly, the prefixes smaller than milli- are potentially useful in measuring extremely small angles.

## Plane Trigonometry Rev Edition

Retrieved February The MacTutor History of Mathematics. The lowest-priced brand-new, unused, unopened, undamaged item in its original packaging where packaging is applicable. Packaging should be the same as what is found in a retail store, unless the item is handmade or was packaged by the manufacturer in non-retail packaging, such as an unprinted box or plastic bag. 