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| Trigonometry |
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Trigonometric
Functions |
Unit of measurement of
angles - a radian (the circular measure) |
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Protractor - an instrument for measuring angles |
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Degrees to radians and radians to degrees conversion examples |
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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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Definitions of
trigonometric functions |
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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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| Unit of measurement of
angles - a radian (the circular measure) |
| The relation between
a central angle a
(the angle between two radii) and the corresponding arc l
in the circle |
| of radius
r is shown by the
proportion, |
| a°
: 360°
= l
: 2rp |
| It shows that the central angle
a°
compared to the round angle of 360° |
| (called
perigon) is in the same relation as the corresponding arc
l |
| compared
to the circumference 2rp.
Therefore, |
 |
|
 |
|
| where the ratio |
 |
we call the
circular measure, usually denoted
arad,
i.e., |
 |
|
| thus, |
 |
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The central angle subtended by the arc equal in length to the radius, i.e.
l = r, |
 |
| we call it
radian. |
| Thus, the angle
a = 1° equals in
radians, |
 |
|
 |
|
| or
arc1° =
0.01745329. Arc is abbreviation from Latin
arcus,
(p =
3.1415926535...). |
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| Protractor
- an instrument for measuring angles |
| Mentioned relations between units of
measurement of an angle and arc clearly shows the protractor
shown in |
| the
below figure marked with radial lines indicating degrees,
radians and
rarely used gradians (the angle of an |
| entire
circle or round angle is 400 gradians). |
 |
| A right angle
equals 100 grad (gradians). |
| The hundredth part of a right angle is
1g
grad,
and one 100th part of 1grad is centesimal arc minute 1c,
and |
| one 100th part of centesimal arc
minute is centesimal arc second 1cc,
therefore |
 |
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| Degrees
to radians and radians to degrees conversion examples |
| Example:
Convert
67° 18´
45" to radians.
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| Solution: The given angle we write in the expanded notation and calculate its decimal
equivalent, |
 |
| then use the formula to convert degrees to radians |
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| Using a scientific calculator, the given conversion can be performed almost direct. |
| Before a calculation
choose right angular measurement (DEG, RAD, GRAD) by pressing
DRG key, then |
|
input, 67.1845
INV ®DEG 67.3125° |
| Because a calculator must use
degrees divided into its decimal part one should press ®DEG
(or ®DD) to |
| get
decimal degrees. Then press
INV DRG® to get
radians, 1.174824753rad. |
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| Example:
Convert 2.785rad
to degrees, minutes and seconds. |
| Solution: Using formula, |
 |
| The same result one obtains with a calculator through the
procedure, press DRG key to set RAD
|
| measurement,
then input 2.785
INV DRG® 177.2986066
grad = 177g29c86cc, |
| press
again INV DRG®
159.5687459° obtained are
decimal degrees (DEG), |
| and to convert to degrees/minutes/seconds press
INV ®DMS to
get 159° 34´
7.48". |
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| Example:
Find the length of the
arc l
that subtends the central angle a
= 123°
38´ 27"
in the circle |
| of
radius r =
15 cm. |
| Solution: First express the angle
a
in decimal degrees, i.e. |
 |
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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
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| 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
|
| 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
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| 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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|
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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. |
 |
| 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
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| circular measure of the
subtended central angle
a as is
shown in the below figure. |
| In accordance with
the definitions of trigonometric functions |
| in a right-angled
triangle, |
| - the sine of an angle
a
(sina)
in a right triangle is the ratio
of the side opposite the angle to the
hypotenuse. |
| - the
cosine of an angle
a
(cosa) in a
right triangle is the ratio of the side adjacent to it to the hypotenuse. |
| Thus, from the
right triangle OP′P,
follows |
| sinx
= PP′
The sine of arc x
is the ordinate of the arc |
|
endpoint. |
| cosx
= OP′ The cosine of arc
x
is the abscissa of the arc |
|
endpoint. |
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| The tangent of an angle
a
(tana)
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 |
 |
| 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, |
| tanx
= 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, |
 |
|
| cotx
= 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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| 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 |
|
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
|
| 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
|
| table.
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| Trigonometry
contents |
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| Copyright
© 2004 - 2012, Nabla Ltd. All rights reserved. |
