Trigonometric Functions:
the trigonometric functions are the functions which relate an angle of a right-angled triangle to ratios of two side lengths. These are also called circular functions, angle functions or goniometric functions.
The most widely used trigonometric functions are the sine, the cosine, and the tangent. Their reciprocals are respectively the cosecant, the secant, and the cotangent, which are less used in modern mathematics. Each of these six trigonometric functions has a corresponding inverse function (called inverse trigonometric function), and an equivalent in the hyperbolic functions as well.
The trigonometric functions of an acute angle \((0\le\theta\le90°)\) are defined in the context of a right triangle as follows:
Sine (sin): It is the ratio of the length of the side that is opposite to that angle, to the length of the longest side (hypotenuse) of the triangle. For an acute angle \(\theta\), the sine function is denoted by \(sin\theta\).
\(sin\theta=\frac{opposite}{hypotenuse\ }=\frac{a}{h}\)
Cosine (cos): It is the ratio of the length of the side that is adjacent to that angle, to the length of the longest side (hypotenuse) of the triangle. For an acute angle \(\theta\), the cosine function is denoted by \(cos\theta\).
\(cos\theta=\frac{adjacent}{hypotenuse\ }=\frac{b}{h}\)
Tangent (tan): It is the ratio of the length of the side that is opposite to that angle, to the length of the side adjacent of the angle. For an acute angle \(\theta\), the cosine function is denoted by \(tan\theta\).
\(tan\theta=\frac{opposite}{adjacent\ }=\frac{a}{b}\)
Reciprocal Functions: for the functions stated above, there are reciprocal functions for each as follows:
Cosecant (cosec): It is the reciprocal of Sine and can be defined as the ratio of the longest side of right triangle to the side opposite to the angle. For an acute angle \(\theta\), the cosecant function is denoted by \(cosec\theta\).
\(cosec\theta=\frac{hypotenuse}{opposite}=\frac{h}{a}\)
Secant (sec): It is the reciprocal of Cosine and can be defined as the ratio of the longest side of right triangle to the side adjacent to the angle. For an acute angle \(\theta\), the secant function is denoted by \(sec\theta\).
\(sec\theta=\frac{hypotenuse}{adjacent}=\frac{h}{b}\)
Cotangent (cot): It is the reciprocal of Tangent and can be defined as the ratio of the side adjacent to the angle to the side opposite to the angle. For an acute angle \(\theta\), the cotangent function is denoted by \(cot\theta\).
\(cot\theta=\frac{adjacent}{opposite}=\frac{b}{a}\)