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| Trigonometry |
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Trigonometric
Equations |
Equations that Can be
Written as f
· g = 0 |
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Trigonometric
Equations of Quadratic Form |
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The
Basic Strategy for Solving Trigonometric Equations |
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| Equations that can be
written as f
· g = 0 |
| If the given equation, by using appropriate transformations, can be rearranged to the form
f
· g = 0 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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that is, 2sin
30° · cos x = 2cos2 x |
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or cos
x -
2cos2 x = 0,
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. |
| Thus, the solution set of the given equation
we can write as |
| {90° + k · 180° : kÎ
Z} U {+
60° + k · 360° : kÎ
Z} or {90° + k · 180°, +
60° + k · 360° : kÎ
Z}. |
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| Trigonometric
equations of quadratic form |
| The trigonometric equation of the quadratic form
[F (x)]2
+ p · F (x) + q = 0, |
| where
F (x)
denotes given trigonometric function, by substituting F
(x) = u becomes a quadratic equation |
 |
| returned into substitutions
F
(x) = u1
and F
(x) = u2
lead to the known basic trigonometric equation. |
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Example: Find the solution set for the equation,
3sin x = 2cos2
x.
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Solution: Using known identity
we write 3sin
x -
2 · (1 -
sin2 x) = 0,
and by plugging
sin x =
u
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| Therefore,
sin x =
1/2,
x = 30° + k · 360°, kÎ Z,
and x′ = 150° + k · 360°, kÎ Z, |
| while the equation
sin x =
-
2 has no solutions since
-
2 is not in the range of the sine function. |
| Thus, the solution set of the given equation is
{30° + k · 360°, 150° + k · 360° : kÎ
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 |
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of the known types. |
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| Pre-calculus contents
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