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Trigonometry |
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Trigonometric
Functions |
The unit circle or
the trigonometric circle |
Division of the circumference of the unit circle to
the characteristic angles |
Definitions of
trigonometric functions |
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The unit circle or
trigonometric circle |
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. |
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. |
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 |
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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Definitions of
trigonometric functions |
Let x
be an arc of the unit circle measured counterclockwise from
the
x-axis. It is at the same time the circular measure of the
subtended central angle
a as is shown in the below
figure
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In accordance with
the definitions of trigonometric functions |
in a right-angled
triangle, |
- the sine of an angle
a
(sin a)
in a right triangle is the ratio
of the side opposite the angle to the
hypotenuse. |
- the
cosine of an angle
a
(cos
a) in a
right triangle is the ratio of the side adjacent to it to the hypotenuse. |
Thus, from the
right triangle OP′P,
follows |
sin x
= PP′
The sine of arc x
is the ordinate of the arc |
endpoint. |
cos x
= OP′ The cosine of arc
x
is the abscissa of the arc |
endpoint. |
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The tangent of an angle
a
(tan a)
in a right triangle is the ratio of the lengths of the opposite to the adjacent side. |
The cotangent is defined as reciprocal of the tangent,
thus |
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From the similarity of the triangles
OP′P
and OP1S1, |
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Hence, the definition of the tangent function in the unit
circle, |
tan
x
= P1S1
The tangent of an arc x
is the ordinate of intersection of the second or terminal side (or its |
extension) of the given angle and the tangent line x =
1. |
From the similarity of the triangles,
OP′P
and OP2S2, |
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cot
x
= P2S2
The cotangent of an arc x
is the abscissa of intersection of the second or terminal side (or its |
extension) of the given angle and the tangent
y
= 1. |
It is obvious from the definitions that the tangent function is not defined for arguments
x
for which cos x =
0,
as well as the cotangent function is not defined for the arguments for which
sin x = 0. |
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Functions
contents B
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