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Trigonometry
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Trigonometric
equations
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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. |
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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 |
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c) = F(a +
k · P)
or bx +
c = a +
k · P,
thus |
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The equation
tan
(bx
+ c) = m example
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Example:
Solve
the equation, |
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Solution:
Rewrite
the equation to the form tan
(bx + c) = m,
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We obtain the same result
if we put given parameters,
b,
c, and corresponding basic solution
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x0
= a
= tan-1
m to the common
solution, i.e., b
= 1/3,
c = -
p/2,
x0
= a
= tan-1(-Ö3/3) =
-
p/6
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The equation
cot
(bx
+ c) = m example
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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
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cot (2x
-
10°) = cot (135° + k · 180°),
2x -
10° = 135° + k · 180° => x = 72°30′ +
k · 90°.
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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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Equations that can be
written as f
· g = 0
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If the given
equation can be rearranged to the form
f
· g = 0
, by using appropriate transformations, then its solution is represented as the union of the individual solutions of the equations
f = 0 and
g = 0.
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Example:
Solve the equation,
sin (x +
30°) + sin
(30° -
x) = 2cos2 x.
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Solution:
Using the sum to product formula (or addition formula)
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| given equation gets the form
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is, 2sin
30° · cos x = 2cos2 x
or
cos
x -
2cos2 x = 0,
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cos x
· (1 -
2cos x) = 0,
the equation of the form f
· g = 0
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therefore,
cos
x = 0,
x = 90° + k · 180°, kÎ Z,
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and 1 -
2cos x = 0,
cos x = 1/2,
x = +
60° + k · 360°, kÎ Z.
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the solution set of the equation we
write, {90° + k · 180° , kÎ
Z} U
{+
60° + k · 360°, kÎ
Z}
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{90° + k · 180°, +
60° + k · 360°, kÎ
Z}.
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Contents F
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Copyright
© 2004 - 2020, Nabla Ltd. All rights reserved.
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