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| Trigonometry |
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Trigonometric
Functions |
Periodicity of
trigonometric functions |
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Signs of trigonometric
functions |
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The table of signs of trigonometric
functions |
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| Periodicity of
trigonometric functions |
| After the argument (arc)
x
passes through all real values from the interval 0
< x < 2p or after the terminal |
| side
of an angle turned round the origin for an entire circle, trigonometric or circular functions will repeat their |
| initial values. |
| As the terminal point
P
of an arc continue rotation around a unit circle in the positive direction passing over |
| the initial point
P1 , it takes next values from the interval
2p
<
x < 4p, then the values from the interval |
| 4p
< x < 6p and so on. |
| On the same way we can examine the rotation of the terminal point
P of an arc
x in the
negative (clockwise) |
| direction, when it will pass through the values from the
intervals, 0 to
-2p, from
-2p
to -4p, and so on. |
| It follows that the argument
x can take any
value, |
| x
= arad
+ k · 2p,
k
= 0, ±1, ±2, ±3,
. . . or
x
= a°
+ k · 360°, k
Î
Z. |
| that is, every real value between
-
oo
and +
oo. |
| Particularly, for
k = 0, i.e., during the first rotation the value of
argument is x
= arad. |
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While the arc endpoint continues rounding over the starting point the trigonometric functions will, in
every |
| interval of length
2p (i.e., from
2p to
4p, from
4p to
6p,
. . . , or from
0 to
-2p, from
-2p to
-4p,
. . . ) |
| take the same values in the same order they took in the first interval
[0,
2p]. |
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| Functions which have the characteristic to take the same values while their argument changes for all integral |
| multiples of a constant interval (or a constant increases in amount called increment) we call
periodic |
| functions, and this constant interval we call
period. |
| Hence, we say that trigonometric functions are periodic functions of
x, so that |
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f (x) = sin x
and f (x)
= cos x of
the period P
= 2p, |
| while
functions, f
(x)
= tan x
and f
(x)
= cot x of
the period P =
p. |
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The periodicity of trigonometric functions show the
identities,
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sin
(a
+ k · 2p)
= sin a
and cos
(a
+ k · 2p)
= cos a,
k
Î Z |
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tan
(a
+ k · p)
= tan a
and cot
(a
+ k · p)
= cot a,
k
Î Z |
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Signs of trigonometric
functions
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Quadrant I
- Values of trigonometric functions, sine, cosine, tangent and cotangent of any arc from
the first
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quadrant are all positive as positive are the coordinates of the points,
P,
S1
and S2
that define their values.
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Quadrant II
- For arcs from the second quadrant points, P
and S2
both have negative abscissas (see the
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above figure), so the cosine and cotangent are negative. The ordinate of the terminal point
P
is positive so
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that the sine is positive while the ordinate of the point
S1
is negative, thus the tangent is negative.
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Quadrant III
- As the abscissas and the ordinates of the terminal points P
of arcs from the third quadrant
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(see the above figure) are negative it follows that cosine and sine functions of these arcs are negative. The
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ordinates of the points S1 and the abscissas of the points
S2 that belong to the arcs from the third quadrant
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are positive. Thus, the tangent and cotangent of these arcs are positive.
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Quadrant IV
- The functions, sine, tangent and cotangent of the arcs from the fourth quadrant are negative
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as are the coordinates of the points, P,
S1
and S2, that belong to them. Only the cosine function of arcs from
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the fourth quadrant is positive as are the abscissas of points
P that belong to them
(see the above figure).
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The table of signs of trigonometric
functions
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| Example:
To which quadrant belongs the endpoint of an arc
a
if
sin a < 0 and
cot a
> 0.
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Solution: The
right figure shows that the ordinate of |
| the endpoint P
of an arc from the third quadrant is |
| negative, so sin
a < 0
while the abscissa of the
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| point (in which the extension of the terminal side of |
| the angle
a
intersects the tangent y =
1) is positive, |
| i.e., cot
a
> 0.
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| That is in accordance with the signs in the above
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| table.
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| Functions
contents A |
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| Copyright
© 2004 - 2012, Nabla Ltd. All rights reserved. |
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