3.3 SolvingTrigonometricEquations
[Pages:4]
Chapter 3. Trigonometric Identities and Equations, Solution Key
3.3 Solving Trigonometric Equations
1. Answer: ? Because the problem deals with 2, the domain values must be doubled, making the domain 0 2 < 4 ? The reference angle is = sin-1 0.6 = 0.6435
? 2 = 0.6435, - 0.6435, 2 + 0.6435, 3 - 0.6435
? 2 = 0.6435, 2.4980, 6.9266, 8.7812
? The values for are needed so the above values must be divided by 2.
? = 0.3218, 1.2490, 3.4633, 4.3906
? The results can readily be checked by graphing the function. The four results are reasonable since they are the only results indicated on the graph that satisfy sin 2 = 0.6.
2.
cos2 x = 1 16
cos2 x =
1
16
cos x = ? 1 4
Then cos x = 1 4
cos-1 1 = x 4
or
cos x = - 1
4
cos-1 - 1 = x 4
x = 1.3181 radians
x = 1.8235 radians
?
However,
cos x
is
also
positive
in
the
fourth
quadrant,
so
the
other
possible
solution
for
cos x =
1 4
is
2 - 1.3181 = 4.9651 radians
?
cos
x
is
also
negative
in
the
third
quadrant,
so
the
other
possible
solution
for
cos
x
=
-
1 4
is 2 - 1.8235 =
4.4597 radians
3.
tan2 x = 1
tan x = ? 1
tan x = ?1
41
3.3. Solving Trigonometric Equations
? So, tan x = 1 or tan x = -1.
?
Therefore, x is all critical values corresponding with
4
within the interval.
x=
4
,
3 4
,
5 4
,
7 4
4. Use factoring by grouping.
2 sin x + 1 = 0 or
2 sin x = -1 sin x = - 1 2 x = 7 , 11 66
2 cos x - 1 = 0
2 cos x = 1 cos x = 1
2 x = , 5
33
5. You can factor this one like a quadratic.
sin2 x - 2 sin x - 3 = 0
(sin x - 3)(sin x + 1) = 0
sin x - 3 = 0
sin x = 3
or
x = sin-1(3)
sin x + 1 = 0 sin x = -1 x = 3 2
For
this
problem
the
only
solution
is
3 2
because
sine
cannot
be
3
(it
is
not
in
the
range).
6.
tan2 x = 3 tan x tan2 x - 3 tan x = 0 tan x(tan x - 3) = 0
tan x = 0 or x = 0,
tan x = 3 x = 1.25
42
Chapter 3. Trigonometric Identities and Equations, Solution Key
7.
2
sin2
x 4
-
3
cos
x 4
=0
2
1 - cos2 x
x - 3 cos = 0
4
4
2 - 2 cos2 x - 3 cos x = 0
4
4
2 cos2 x + 3 cos x - 2 = 0
4
4
x
x
2 cos - 1 cos + 2 = 0
4
4
x 2 cos - 1 = 0 or
4 x
2 cos = 1 4
x1 cos =
42
x = or 5
43
3
x = 4 or 20
3
3
x cos + 2 = 0
4 x
cos = -2 4
20 3
is
eliminated
as
a
solution
because
it
is
outside
of
the
range
and
cos
x 4
=
-2
will
not
generate
any
solutions
because
-2
is
outside
of
the
range
of
cosine.
Therefore,
the
only
solution
is
4 3
.
8.
3 - 3 sin2 x = 8 sin x
3 - 3 sin2 x - 8 sin x = 0
3 sin2 x + 8 sin x - 3 = 0
(3 sin x - 1)(sin x + 3) = 0
3 sin x - 1 = 0 or sin x + 3 = 0
3 sin x = 1
sin x = 1 3
sin x = -3
x = 0.3398 radians No solution exists
x = - 0.3398 = 2.8018 radians
9. 2 sin x tan x = tan x + sec x
sin x sin x 1 2 sin x ? = +
cos x cos x cos x 2 sin2 x sin x + 1
= cos x cos x 2 sin2 x = sin x + 1
2 sin2 x - sin x - 1 = 0
(2 sin x + 1)(sin x - 1) = 0
2 sin x + 1 = 0
or
2 sin x = -1
1 sin x = -
2
sin x - 1 = 0 sin x = 1
x = 7 , 11 66
43
3.3. Solving Trigonometric Equations
One
of
the
solutions
is
not
2
,
because
tan x
and
sec x
in
the
original
equation
are
undefined
for
this
value
of
x.
10.
11. tan2 x + tan x - 2 = 0
2 cos2 x + 3 sin x - 3 = 0
2(1 - sin2 x) + 3 sin x - 3 = 0 Pythagorean Identity
2 - 2 sin2 x + 3 sin x - 3 = 0
- 2 sin2 x + 3 sin x - 1 = 0 Multiply by - 1
2 sin2 x - 3 sin x + 1 = 0
(2 sin x - 1)(sin x - 1) = 0
2 sin x - 1 = 0
or sin x - 1 = 0
2 sin x = 1
sin x = 1 2
x = , 5 66
sin x = 1 x= 2
-1 ?
12 - 4(1)(-2) = tan x
2 -1 ? 1 + 8
= tan x 2 -1 ? 3
= tan x 2
tan x = -2 or 1
tan x
=
1
when
x
=
4
,
in
the
interval
-
2
,
2
tan x = -2 when x = -1.107 rad
12. 5 cos2 - 6 sin = 0 over the interval [0, 2].
5 1 - sin2 x - 6 sin x = 0 -5 sin2 x - 6 sin x + 5 = 0
5 sin2 x + 6 sin x - 5 = 0
-6 ? 62 - 4(5)(-5) = sin x
2(5)
-6 ? 36 + 100 = sin x
10 -6 ? 136
= sin x 10 -6 ? 2 34
= sin x 10 -3 ? 34
= sin x 5
x = sin-1
-3+ 34
5
or sin-1
-3- 34
5
x = 0.6018 rad or 2.5398 rad from the first expression, the second
expression will not yield any answers because it is out the the range of sine.
44
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