(Section 5.3: Solving Trig Equations)
5.22
SECTION 5.3: SOLVING TRIG EQUATIONS
PART A: BASIC EQUATIONS IN sin, cos, csc, OR sec (LINEAR FORMS)
Example
Solve:
5cos
x
−
2
=
3cos
x
(It is assumed that you are to give
all
real solutions and to give them in
exact
form
– no approximations – unless otherwise specified.)
Conditional Equations
This is an example of a conditional equation. It is true (i.e., the left side
equals the right side) for some real values of
x
but not for others. In other
words, the truth of the equation is conditional, depending on the particular
real value that
x
takes on. You should be used to solving conditional
equations in your Algebra courses.
This is different from an identity, which holds true for
all
real values of
x
(for instance) for which all expressions involved are defined as real
quantities. An identity may be thought of as an equation that has as its
solution set the intersection (overlap) of the domains of the expressions
involved.
Solution
First
, solve for
cos
x
. This process is no different from solving the linear
equation
5
u
−
2
=
3
u
for
u
. In fact, you could employ the substitution
u
=
cos
x
and do exactly that.
5cos
x
−
2
=
3cos
x
2cos
x
=
2
cos
x
=
2
2
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(Section 5.3: Solving Trig Equations)
5.23
We want to find
all
angles whose cos value is
2
2
. We will use radian
measure, which corresponds to “real number” solutions for
x
.
Second
, because
cos
x
has period
2
π
, we will first find solutions in the
interval
0, 2
π
⎡
⎣
)
. Later, we will find all of their coterminal “twin” angles.
If you are more comfortable with “slightly negative” Quadrant IV angles
such as
−
π
4
than angles such as
7
π
4
, then you may want to look in the
interval
−
π
2
,
3
π
2
⎡
⎣
⎢
⎞
⎠
⎟
, instead.
Is there an “easy” angle
x
whose cos value is
2
2
?
Yes, namely
π
4
, which is
cos
−
1
2
2
⎛
⎝
⎜
⎞
⎠
⎟
.
Look at the Unit Circle. Look at the point corresponding to the
π
4
angle.
It turns out that there is another point on the Unit Circle that has the same
horizontal (or what we used to call “
x
”) coordinate,
2
2
, so we must look for
another angle with that same cos value of
2
2
. We know that this point lies
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 Fall '11
 staff
 Calculus, Trigonometry, Equations, Unit Circle, trig equations, Cos

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