This calculator uses a special trigonometric rule to demonstrate the Law of Sines, as follows: Given a triangle of sides A-B-C and angles of a-b-c, where complementary letters are the side and angle opposite each other, *A / sin(a) = B / sin(b) = C / sin(c)*.
The ratios between all three pairs of sides and angles will always be the same, regardless of what shape or size of triangle. This applies to right triangles, 30-60-90 triangles, Isosceles triangles, equilateral triangles, and every other three-sided geometric shape you can draw on a flat plane.
The **Law of Sines** is thus used to compute triangle parameters when we are given incomplete information. This process is called "triangulation."
One can find the remaining sides and angles of a triangle given two angles and one side, for instance. In our calculator, entering any three values causes it to output the third.

The three ratios of angle sines to sides also the diameter of the same triangle's circumcircle, again no matter what the shape of the triangle. All triangles have a circumcircle, a fact which we can prove simply by drawing any circle and noting that any three points we indicate along that circle's edge form a triangle. This property was first pointed out by Greco-Roman philosopher Ptolemy (100 A.D.).
There are ambiguous cases of triangles where not enough information is provided to arrive at a correct answer, but instead, two answers are possible. The conditions are:
**•** Only one angle, **a**, is known.
**•** Only two sides, **A** and **C**, are known.
**•** Angle **a** is acute, meaning it's less than 90 degrees.
**•** Side **A** is shorter than side **C**.
In this case, two triangles can be constructed from this information: either a right triangle or an irregular triangle formed of a right triangle and an equilateral triangle with congruent height.

Sines are one of a set of three trigonometric functions, together with cosines and tangents. In labeling a right triangle with ABC for the sides and abc for the angles, such that angle c is 90 degrees, we have:
**•** *sin (a) = (A / C)*
**•** *cos (a) = (B / C)*
**•** *tan (a) = (A / B)*
When we understand how to use trigonometric functions, we can compute and manipulate waves of any kind, be they shock waves, ocean waves, light waves, or sound waves.
In fact, sound engineers have tones they produce directly with labels "**sin**," "**cos**," and "**tan**." Graphing calculators are a popular tool for those interested in further investigation of trigonometric wave functions.