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| Trigonometry |
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Trigonometric
Functions |
The unit circle or
the trigonometric circle |
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Division of the circumference of the unit circle to
the characteristic angles |
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| The unit circle or
trigonometric circle |
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A circle of radius r =
1, with the center at the origin
O(0, 0) of a coordinate system, we call the
unit or trigonometric
circle, see the figure below. |
| The arc of the unit circle that describes a point traveling
anticlockwise
(by convention, clockwise is taken to be negative direction) from the initial position
P1(1,
0) on the x-axis, along
the circumference, to the terminal position P
equals the angular measure/distance x
= arad,
in radians.
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| An angle is in standard position if its initial side lies along the
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| positive
x-axis. |
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If we take the positive direction of the x-axis as the beginning
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| of a measurement of an angle (i.e.,
a
= 0rad,
both sides of
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| an angle lie on the
x-axis), and the unit point
P1
as the initial
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| point of measuring the arc, then the terminal side of an angle,
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| which passes through the terminal point
P
of the arc,
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rotating around the origin
(in any direction) describes different |
| angles,
and the terminal point P
corresponding arcs, |
| x
= arad
+ k · 2p,
k
= 0,
±1,
±2,
±3,
. . . |
| or
x
= a°
+ k · 360°, k
Î
Z. |
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It means that every arc x
ends in the same point P
in which ends the corresponding arc
a.
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| Thus, at any
point P
on the circumference of the unit circle end infinite arcs
x = a
+ k ·
2p,
which differ by the
multiplier
2p, and any number
x associates only one point
P. |
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| Division of the circumference of the unit circle to
the characteristic angles |
| There
is a common division of the circumference of the unit circle to
the characteristic angles or the corresponding
arcs which are the multipliers of the angles, 30°
(p/
6) and 45°
(p/
4). |
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We can say that a unit circle is at the same time |
| numerical
circle. |
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The numerical circle shown in the right figure is formed
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| by winding the positive part of number line, with the unit |
| that equals the radius,
around the unit circle in the |
| anticlockwise
direction and its
negative part in clockwise |
| direction. |
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| Therefore, the terms angle, arc and number in the
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| trigonometric
definitions and expressions are mutually
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| interchangeable.
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| Example:
In which quadrant lies second or the terminal side
of the angle x
= 1280°. |
| Solution: Dividing the given angle by
360°
we calculate the |
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number of rotations, or round angles, described by terminal
side |
| of the angle x, and the remaining angle
a° position of
which we |
| want to find. |
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| since
x
= a°
+ k · 360°
then
k = 3
and a
= 200°. |
| therefore, terminal side of the angle
x lies in the third quadrant. |
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| Example:
In which quadrant lies the endpoint of the arc
x
= -
47p/3
of a unit circle. |
| Solution: Given
arc |
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| can be expanded to |
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| Thus, the endpoint of the arc
x
moved around a unit circle in the clockwise (negative) direction 7 times and described additional arc
a
= -
(5/3)prad,
so its endpoint P
lies in the first quadrant. |
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| Pre-calculus
contents C |
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| Copyright
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