# 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.

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.