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Trigonometry
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Trigonometric
equations
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| The equations,
sin
(bx + c) = m, -1
<
m <
1,
cos
(bx + c) = m, -1
<
m <
1, |
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tan
(bx
+ c) = m and cot
(bx
+ c) = m,
where
b,
c
and m
are real numbers. |
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The given equations can be written as F(bx +
c) = m
where F
substitutes a trigonometric function, x
is an arc
to be calculated and m
is a value of a given trigonometric function. |
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To every trigonometric function we can determine an arc, a +
k · P of which function value equals
m
that is F(a +
k · P) = m,
where a
= x0
is the basic solution, and P
is the period, then |
| F(bx +
c) = F(a +
k · P)
or bx +
c = a +
k · P,
thus |
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Thus, from obtained general solution we can write a common solutions for every given
equation,
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The equation
sin
(bx + c) = m, -1
<
m <
1 example
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Example:
Solve
the equation, |
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Solution:
Rewrite
the equation to the form sin
(bx + c) = m,
so sin
(2x + p/6)
= -
1/2
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An alternative but similar solution can be obtained by substituting the values of,
b,
c
and m,
into |
| x0
= a
and x′0
= p
-
a
and to the common solution
written above |
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The equation
cos
(bx + c) = m,
-1
<
m <
1, example
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Example:
Find
the solutions of the equation, 2cos
(4x -
30°) + Ö3
= 0.
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| Solution:
Rewrite
the equation to the form cos
(bx + c) = m,
that
is cos
(4 x -
30°) = -
Ö3/2
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| it
follows that cos
(4 x -
30°)
= cos (+ 150° +
k · 360°)
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and
4 x -
30°
= + 150° +
k · 360°
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| therefore,
x = 45° +
k · 90°
and x′ = -
30° +
k · 90°, kÎ
Z.
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The same results we get by substituting the values, b
= 4, c = -
30°
and m =
-
Ö3/2,
into
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| x0
= a
= cos-1
m
= cos-1
(-
Ö3/2)
= 150° and
x′0
= -
a
= -150°
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| then,
using the common solution formulas obtained are
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Contents F
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Copyright
© 2004 - 2020, Nabla Ltd. All rights reserved.
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