Trigonometry as computational geometry

Trigonometry began as the computational component of geometry. For instance, one statement of plane geometry states that a triangle is determined by a side and two angles.

In other words, given one side of a triangle and two angles in the triangle, then the other two sides and the remaining angle are determined. Trigonometry includes the methods for computing those other two sides. The remaining angle is easy to find since the sum of the three angles equals 180 degrees (usually written 180°).

Angle measurement and tables

If there is anything that distinguishes trigonometry from the rest of geometry, it is that trig depends on angle measurement and quantities determined by the measure of an angle. Of course, all of geometry depends on treating angles as quantities, but in the rest of geometry, angles aren’t measured, they’re just compared or added or subtracted.

Trigonometric functions such as sine, cosine, and tangent are used in computations in trigonometry. These functions relate measurements of angles to measurements of associated straight lines as described later in this short course.

Trig functions are not easy to compute like polynomials are. So much time goes into computing them in ancient times that tables were made for their values. Even with tables, using trig functions takes time because any use of a trig function involves at least one multiplication or division, and, when several digits are involved, even multiplication and division are slow.

In the early 17th century computation sped up with the invention of logarithms and soon after slide rules. With the advent of calculators computation has become easy. Tables, logarithms, and slide rules aren’t needed in trigonometric computations.

All you have to do is enter the numbers and push a few buttons to get the answer. One of the things that used to make learning trig difficult was performing the computations.

That’s not a problem anymore!

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