Trigonometric functions of an 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 ,
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.