What Is Trigonometry?

We learn from The Words of Mathematics:

trigonometry (noun), trigonometric (adjective): the first part of the word is from Greek trigon "triangle". The second part of trigonometry is from Greek metron "a measure." The Indo-European root is probably me- "to measure." Trigonometry is literally the measuring (of angles and sides) of triangles. Historically speaking, the triangular approach to trigonometry is ancient, wheres the circular approach now taught in our schools is relatively recent.

So originally trigonometry was understood to define relations between elements of a triangle. In a triangle, there are six basic elements: 3 sides and 3 angles. Not any three line segments may serve as the sides of a triangle. They do iff they satisfy the triangle inequality, or rather three triangle inequalities. Not any three angles may be the angles of a triangle. In Euclidean geometry, the three angles of a triangle add up to a straight angle. These requirements impose limitations on the manner in which the relations between the elements are defined. In modern trigonometry these relations are extended to arbitrary angles. This can be done, for example, by observing the projections of a rotating radius of a circle and a tangent at the end of the radius.

If the sides a, b, c of a triangle lie opposite angles α, β, γ then a + b > c is oneof the inequalities that the sides obey, and α + β + γ = 180° is the identity that holds in Euclidean geometry. We also know that, if γ is right, then the Pythagorean theorem holds: a² + b² = c². (Its converse holds too.) Trigonometric relations involve trigonometric functions.

There is an awful amount of trigonometric identities. The most basic one is the Pythagorean theorem expressed in terms of sine and cosine:

sin² α + cos² α = 1.

Then there are double argument formulas:

sin 2α = 2 sin α cos α
cos 2α = cos² α - sin² α
tan 2α = 2 tan α / (1 - tan² α)
cot 2α = (cot² α - 1) / 2cot α.

and, more general, sum and difference formulas:

sin (α + β) = sin α cos β + cos α sin β
cos (α + β) = cos α cos β - sin α sin β
sin (α - β) = sin α cos β - cos α sin β
cos (α - β) = cos α cos β + sin α sin β.

And, of course, no list of trigonometric relations could be complete unless the Laws of Cosines and Sines are mentioned.

Trigonometry is a methodology for finding some unknown elements of a triangle (or other geometric shapes) provided the data includes a sufficient amount of linear and angular measurements to define a shape uniquely. For example, two sides a and b of a triangle and the angle they include define the triangle uniquely. The third side c can then be found from the Law of Cosines while the angles α and β are determined from the Law of Sines. The latter can be used to find the circumradius. The area of the triangle can be found from S = (ab sin γ)/2 and knowing that we can determine the inradius from S = (a + b + c)r/2, and so on.

References

I. M. Gelfand, M. Saul, Trigonometry, Birkhäuser, 2001
S. Schwartzman, The Words of Mathematics, MAA, 1994

Trigonometry

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