Download Do 7.6

Chapter 7
Section 7.6
Trigonometric Equations
If a more complicated angle is inside
the sine or cosine there is one more
step of solving for x at the end.
The example to the right is how to
solve:
i.e. c  
 2
2
cos(5 x ) 
Solutions:
5x 
3
 2k
4
and
5x 
5
 2k
4
 2
2
5x  135
1. Draw unit circle.
cos(5x)   2 2
2. Draw horizontal or vertical line the
correct distance on x or y axis.
3
4
2
2
3. Find angles where line hits the
unit circle.
 2
2
4. Add 2k to each angle.
5
4

5. Solve for x.
 k  360 and 5 x  225  k  360
3 2k
5 2k

and x 

x  27   k  72 and x  45  k  72
20
5
20
5
The Equations: a sinx + b = c and a cosx + b = c

x


If the sine or cosine is not isolated (i.e. all by itself on one side of the equation) carry out the algebra to
isolate the sine or cosine.
Now apply what we did above to get the solutions.
2 cos(5 x)  2  0
Solve:
2 cos(5 x)  2  0
2 cos(5 x)   2
cos(5 x) 
 2
2
Solutions:
x
3 2k

20
5
and
x
5 2k

20
5
x  27

 k  72 and x  45  k  72

Give answers in
radians.
Regroup:
Simplifying
Equations
Before Solving
Sometimes
equations
might need to
be simplified
using a
combination of
trigonometric
identities and
algebra before
solving them.
Factor:
Factor:
2 sin 𝜃 − 1 2 cos 𝜃 + 1 = 0
Solve each equation
2𝜋
3
−1
2
2 cos 𝜃 + 1 = 0
2 cos 𝜃 = −1
−1
cos 𝜃 =
2
𝜃 = 2𝜋
+ 2𝑘𝜋
3
𝜃 = 4𝜋
3 + 2𝑘𝜋
4𝜋
3
Solve:
2 sin x  sin x
2
2 sin x  sin x  0
sin x2 sin x  1  0
Equations
with Powers
of Sine or
Cosine
2
sin x  0 and
If the equation
you are trying
to solve has a
power of sine
or cosine, set
one side
equal to zero
and factor the
other side.
Use what was
just discussed
to solve the
parts you get.
Solve:
2 sin x  1  0
sin x  0 and
sin x 
5
6

cos 2  2x   4 cos 2x   3  0
cos 2x   3cos 2x   1  0
cos 2x   3  0 and cos 2x   1  0
cos 2x   3 and cos 2x   1
1
2

6
0
No Solutions
(first equation)

Solutions:
Solutions:
   2k
x  0  2k
and
x    2k
x
2
x  6  2k
and
x  56  2k
x  2  4k
Rearrange
Square both sides
Regroup
Apply Identity
Cancel
Solve:
5 sin 2 x  14 cos x  13  0
Equations With
Both Sine and
Cosine
5(1  cos 2 x)  14 cos x  13  0
If a
trigonometric
equation has
both a sine
and cosine in it
use
trigonometric
identities to
change it to an
equation
involving either
all sine or all
cosine.
 5 cos x  14 cos x  8  0
36.8699
5  5 cos 2 x  14 cos x  13  0
323.13
2
5 cos x  14 cos x  8  0
(cos x  2)(5 cos x  4)  0
2
cos x  2  0 and
cos x  2 and
No Solution and
Find the other angle.
360-36.8699=323.13
5 cos x  4  0
cos x 
4
5
x  cos
1 4
5
The solutions for this are:

x  36.8699

x  36.8699  k  360

x  323.13  k  360


