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| Trigonometry |
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Trigonometric
Equations |
Homogeneous Equations
in sin
x and cos
x |
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Homogeneous equations
of first degree a×sin x
+ b×cos
x = 0 |
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Homogeneous equations
of second degree a
sin2
x
+ b sin x
· cos
x + c
cos2
x = 0
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The
basic strategy for solving trigonometric equations |
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| Homogeneous equations
in sin
x and cos
x |
| An equation is said to be homogeneous if all its terms are of the same degree.
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| Homogeneous equations
of first degree a sin x
+ b
cos
x = 0 |
| Divide given equation by
cos x to obtain |
| a
tan x + b =
0
or tan
x = -
b/a
the basic equation
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| whose solution is
x =
tan-1
(-
b/a) + k · p
or x
= arctan
(-
b/a) + k · p,
k Î Z. |
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Example: Solve the equation,
-
sin x
+
Ö3
· cos x = 0.
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Solution: Dividing given equation by
-
cos x
obtained is
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| tan
x =
Ö3,
x =
tan-1Ö3
+ k · p =
p/3
+ k · p,
k Î Z. |
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| Homogeneous equations
of second degree a
sin2
x
+ b sin x
· cos
x + c
cos2
x = 0 |
| After division of the given equation by
cos2
x obtained is quadratic
equation |
| a
tan2
x
+ b tan x + c = 0 |
| which is
explained in the previous section. |
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| The
next example shows that the equation a
sin2
x
+ b sin x
· cos
x + c
cos2
x = d
is also homogeneous. |
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Example: Find the solution set for the equation,
5
sin2
x
+ sin x · cos
x + 2
cos2
x = 4.
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Solution: Given equation is equivalent to the equation
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| 5
sin2
x
+ sin x · cos
x + 2
cos2
x = 4
· (sin2
x
+ cos2
x),
since sin2
x
+ cos2
x = 1 |
| which,
when simplified becomes |
| sin2
x
+ sin x · cos
x - 2
cos2
x = 0
- the homogeneous equation of the second degree. |
| Division by cos2
x
gives, |
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thus, (tan x)1
= - 2,
x =
tan-1(- 2)
= -
63°26′05″ + k · 180°,
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(tan x)2
= 1,
x′ =
tan-1
1
= 45° + k · 180°,
k Î Z. |
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| The
basic strategy for solving trigonometric equations |
| When solving trigonometric equations we usually use some of the following
procedures, |
| - apply known
identities to modify given equation to an equivalent expressed in terms of one function, |
| - rearrange the given
equation using different trigonometric formulae to an equivalent, until you recognize one
of the known types. |
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| Pre-calculus contents
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