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,
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.