### Theory [Trigonometric Ratios and Inverse Trigonometric Functions] Trigonometric ratios are the ratios between edges of a right triangle. Greek mathematician Hipparchus is regarded as the Father of Trigonometry, as he was the one who created the first Trigonometric Table. These ratios depend only on one acute angle of the right triangle, since any two right triangles with the same acute angle are similar. So, these ratios define functions of this angle that are called trigonometric functions. Explicitly, they are defined below as functions of the known angle A, where \(a\), \(b\) and \(h\) refer to the lengths of the sides in the accompanying figure:
Sine (denoted sin), defined as the ratio of the side opposite the angle to the hypotenuse.
$$\sin A=\frac {\textrm {opposite}}{\textrm {hypotenuse}}=\frac {a}{h}$$
Cosine (denoted cos), defined as the ratio of the adjacent leg (the side of the triangle joining the angle to the right angle) to the hypotenuse.
$$\cos A=\frac {\textrm {adjacent}}{\textrm {hypotenuse}}=\frac {b}{h}$$
Tangent (denoted tan), defined as the ratio of the opposite leg to the adjacent leg.
$$\tan A=\frac {\textrm {opposite}}{\textrm {adjacent}}=\frac {a}{b}$$
The hypotenuse is the side opposite to the \(90^{\circ}\) angle in a right triangle; it is the longest side of the triangle and one of the two sides adjacent to angle A. The adjacent leg is the other side that is adjacent to angle A. The opposite side is the side that is opposite to angle A. The terms perpendicular and base are sometimes used for the opposite and adjacent sides respectively.
The reciprocals of these ratios are named the cosecant (csc), secant (sec), and cotangent (cot), respectively:
$$\csc A=\frac {\textrm {hypotenuse}}{\textrm {opposite}}=\frac {h}{a}$$
$$\sec A=\frac {\textrm {hypotenus}}{\textrm {adjacent}}=\frac {h}{b}$$
$$\cot A=\frac {\textrm {adjacent}}{\textrm {opposite}}=\frac {b}{a}$$
The cosine, cotangent, and cosecant are so named because they are respectively the sine, tangent, and secant of the complementary angle abbreviated to "co-". |