Trigonometric functions of an acute angle

Trigonometric functions of acute angle: 
sine, cosine, tangent, cotangent, secant, cosecant. 
Exact values of trigonometric functions
for some most used acute angles.

Trigonometric functions of an acute angle are ratios of different pairs of sides of a right-angled triangle ( Fig.2 ).

   1)   Sine:  sin A = a / c   ( a ratio of an opposite leg o a hypotenuse ) .  

   2)   Cosine:  cos A = b / c  ( a ratio of an adjacent leg to a hypotenuse ) .

   3)   Tangent:  tan A = a / b   ( a ratio of an opposite leg to an adjacent leg ) .

   4)   Cotangent:  cot A = b / a   ( a ratio of an adjacent leg to an opposite leg ) .

   5)   Secant:   sec A = c / b   ( a ratio of a hypotenuse to an adjacent leg ) .

   6)  Cosecant:  cosec A = c / a   ( a ratio of a hypotenuse to an opposite leg ) .

There are analogous formulas for another acute angle B ( Write them, please ! ).

E x a m p l e .  A right-angled triangle ABC  ( Fig.2 ) has the following legs:
                       a = 4,  b = 3. Find sine, cosine and tangent of angle A.

S o l u t i o n .  At first we find a hypotenuse, using Pythagorean theorem:

                       c ² = a ² + b ², 

                       According to the above mentioned formulas we have:
                       sin A = a / c = 4 / 5;  cos A = b / c = 3 / 5;  tan A = a / b = 4 / 3. 

For some angles it is possible to write exact values of their trigonometric functions. The most important cases are presented in the table:

Although angles 0° and 90° cannot be acute in a right-angled triangle, but at enlargement of notion of trigonometric functions ( see below), also these angles are considered. A symbol  ∞  in the table means that absolute value of the function increases unboundedly, if the angle approaches the shown value.