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Trigonometry |
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Trigonometric
Equations |
The Equations,
tan
(bx
+ c) = m and cot
(bx
+ c) = m,
where
b,
c and
m are real
numbers.
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The Equation,
cot
(bx
+ c) = m example |
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The Equations,
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. |
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 b
x +
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,
cot
(bx
+ c) = m example |
Example: Find
the solutions of the equation, cot
(-
2x + 10°) -
1 = 0.
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Solution: Rearrange
the given equation to the form cot
(bx + c) = m, thus
cot (-
2x + 10°) = 1,
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or cot
[-
(2x -
10°)] = 1
and since cot
(-
a)
= -
cot a
then, cot
(2x -
10°) = -
1, |
and
cot (2x
-
10°) = cot (135° + k · 180°),
2x -
10° = 135° + k · 180° => x = 72°30′ +
k · 90°. |
The general solution of the equation we get direct substituting the basic solution
x0
= a
and the constant b
and
c
to the common solution, b
= 2, c = -
10° and
x0
= a
= cot-1(-1) =
135°
give |
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Pre-calculus contents
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