Section 5.3, Solving Trigonometric Equations
[Pages:2]Section 5.3, Solving Trigonometric Equations
Homework: 5.3 #7?39 odds
In Section 4.7, we learned about inverse trigonometric functions, which gave only one solution to
equations
like
sin x
=
1 2
.
Now,
we
will
focus
on
finding
all
angles
that
solve
trigonometric
equations.
Examples Solve the following equations:
1. 2 cos x - 1 = 0
2 cos x - 1 = 0
2 cos x = 1
1
cos x =
2
x = + 2n, - + 2n
3
3
2. tan2 x + 3 tan x = 0
We can use techniques from factoring, so:
tan2 x + 3 tan x = 0
tan x(tan x + 3) = 0
tan x = 0
x = n
tan x = - 3
x = - + n
3
3. 4 sin2 x - 1 = 0
4 sin2 x - 1 = 0
(2 sin x - 1)(2 sin x + 1) = 0
1 sin x =
2
5
x = + 2n, + 2n
6
6
1 sin x = -
2
7
11
x = + 2n, + 2n
6
6
4. 3 sin2 x = cos2 x
Here, we have more than one trigonometric function, so we want to change this to having only one trigonometric function. Let's use the identity cos2 x = 1 - sin2 x
3 sin2 x = cos2 x 3 sin2 x = 1 - sin2 x 4 sin2 x - 1 = 0
This
is
now
the
same
as
the
last
problem,
so
x
=
6
+ 2n,
5 6
+ 2n,
7 6
+ 2n,
11 6
+ 2n.
You could also solve this by first dividing both sides by cos2 x, which would give you 3 tan2 x =
1.
5. cot2 4x = 3
cot2 4x = 3
cot 4x = ? 3
4x = + n, - + n
6
6
n n
x= + ,- +
24 4 24 4
6. 1 + tan2 = sec2 All values of will satisfy this equation. This will happen whenever your equation is a trigonometric identity.
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