Trigonometric
functions of arcs from
0
to ± 2p
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Values of
trigonometric functions of arcs
p/6, p/4 and p/3
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The given arcs are, one twelfth, one eighth and one sixth of the circumference
2p
of the unit circle so the coordinates of terminal points of the arcs are the elements of the equilateral triangle with the side
a = 1
(Fig.a, c) and the sides of the square with diagonal
d = 1, (Fig.
b).
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The values of the trigonometric functions of arcs that are multipliers of
30°
(p/6)
and 45°
(p/4)
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Calculation of values of trigonometric functions
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Trigonometric
reduction formulas - the reference angles
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Values of trigonometric functions, of any angle greater than
90° (p/2)
can be expressed by the corresponding value of the function of an angle from the first quadrant.
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The acute angle
a between the
x-axis and the terminal side of angle
x
we call the reference angle.
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If a given arc x
is greater than
2p then, before calculating values of
functions sine and cosine, we should divide it by
2p,
while for functions tangent and cotangent by
p, to determine integral multiplier
k and
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reminder a
therefore,
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and for functions, tangent and cotangent
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These
properties of trigonometric functions are included in the reduction formulas that give the value of
any angle x
greater than 90° (p/2)
in terms of same function of an acute angle a.
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We calculate values of trigonometric functions of an arbitrary angle
x
by using its reference angle
a.
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If x
is an angle in standard position, then its reference angle is given by the acute angle
x
which is enclosed between the terminal side of the
x
and the
x-axis.
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Calculation of values of trigonometric functions of an arbitrary angle
x, examples
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Example:
Calculate sin
1110°.
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Solution: To use the expression x =
a +
k · 360° we should divide given angle by
360°
to get an integer multiple k and remaining angle
a,
thus
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Example:
Calculate
cos
(-
77p/4).
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Solution: Since cos
(-
x) = cos x we can write
cos
(-
77p/4)
= cos 77p/4,
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then, divide given x
by
2p,
x = 77p/4
= 19p
+ p/4
= 9 · 2p
+ 5p/4
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so that,
cos
(-
77p/4)
= cos 77p/4
= cos (9 · 2p
+ 5p/4)
= cos 5p/4.
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As the terminal side of the angle 5p/4
lies in the third quadrant we use cos
(p
+ a) = -
cos a
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therefore,
cos
5p/4
= cos
(p
+ p/4) =
-
cos p/4
= -
Ö2/2.
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Basic relationships
between trigonometric functions of the same angle
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Using the Pythagorean theorem for the right triangle
in the figure we get the fundamental trigonometric identity
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sin2
x
+ cos2 x = 1 |
which, because of periodicity of the trigonometric functions, holds for an arbitrary angle
x =
a +
k · 2p
therefore,
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sin2
x
+ cos2 x = 1,
x
Î
R.
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From this identity and the definitions of the functions, tangent
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and cotangent
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we can derive twelve formulas so that each function is expressed
through another three.
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Basic relationships
between trigonometric functions of the same angle shown in the
unit circle
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Relations between the trigonometric functions of the same angle, expressed by absolute value,
are included in the definitions of the functions in right triangles shown in
Fig. a - d, and included in the table below.
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Basic relationships
between trigonometric functions shown in the tabular form
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Trigonometric
expressions examples
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Example:
Find values of other trigonometric functions of
an angle a,
if given sin a
= -
4/5 and
270° < a
< 360°.
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Solution: Since a
is a forth-quadrant angle, then
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Example:
Find the value
of |
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Solution: Dividing the numerator and
the denominator by
cos2 x,
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