Dictionary.com defines trigonometry as “The branch of mathematics that deals with the relations between the sides and angles of plane or spherical triangles, and the calculations based on them.” The ancient Egyptians and ancient Babylonians were among the first to use a primitive form of trigonometry. The beginning of the concepts of trigonometry today started when the ancient Greek mathematician Hipparchus started using chords to measure circles.
Trigonometry is the branch of mathematics that uses the relationships between the angles and sides of triangles. A triangle is formed by any three connecting segments. These three connecting segments form three sides, three angles, and three points.
The trigonometric functions are cosine, sine, tangent, cotangent, secant, and cosecant use the three points, angles, and sides to define the relationships. The terminology “opposite” and “adjacent” are used at each of the three points to describe the relationships of the functions. Also, one of the angles must be 90 degrees (called a right triangle) and is called the hypotenuse. Since all triangles in plane geometry have 180 degrees, the other two angles always add up to 90 degrees.
The sine is the opposite side divided by the hypotenuse at each of the three points of the triangle. The cosine is the adjacent side divided by the hypotenuse. The tangent is the opposite side divided by the adjacent side. The cotangent is the reciprocal of the tangent—the adjacent side divided by the opposite side. The secant is the reciprocal of the cosine—the hypotenuse divided by the adjacent side. The secant is the reciprocal of the sine—the hypotenuse divided by the opposite side.
The “co” in the cosine, cotangent, and cosecant is complementary abbreviated. This is based on the fact that the sine, tangent, and secant of any angle are equal to the cosine, cotangent, and cosecant of the complement of that angle respectively. The complement of an angle is 90 degrees minus that angle.
The law of sines and the law of cosines can be used for any arbitrary triangle, whether it is a right triangle or not. The law of sines labels the three sides a, b, and c and uses the sines of the angles formed at these three points. The law of sines is a/sin(a)=b/sin(b)=c/sin(c). The reciprocal of the equations is also true: sin(a)/a=sin(b)/b=sin(c)/c. It is used when two angles and one side are known or two sides and one angle not formed by the two sides are known.
The law of cosines uses the sides a, b, c and the angle C opposite side c. The formula is c^2=a^2+b^2-2abC. It is also true, using B for the angle opposite side B and A for the angle opposite side A that b^2=a^2+c^2-2acB and a^2=b^2+c^2-2bcA.
