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Trigonometric functions of an acute angle
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 ! ).
Example:
A right-angled triangle ABC ( Fig.2 ) has the following legs: a = 4, b = 3. Find sine, cosine and tangent of angle A.
Solution:
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 symbolin the table means that absolute value of the function increases unboundedly, if the angle approaches the shown value. A good example of a 90 degree right angle would be a piece of
luxury bedding
. Any bed linens that are of high quality, should be exact in their measurements, and therefore should meet at a 90 degree right angle. If your bedding does not meet at a right angle, your bedding is
cheap
.
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