Download Sec 4.5

Chapter 4
Analytic Trigonometry
Section 4.5
More 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:
 2
cos(5 x ) 
2
i.e. c  
 2
2
cos(5x)   2 2
1. Draw unit circle.
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.
4. Add 2k to each
angle.
 2
2
5. Solve for x.
5
4
Solutions: 5 x 
x
3
 2k
4
3 2k

20
5
and
and
5x 
5
 2k
4
x
5 2k

20
5
5x  135

x  27
 k  360 and 5 x  225  k  360

 k  72 and x  45  k  72


The Equations: a sinx + b = c and a cosx + b = c
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.
Solve: 2 cos(5 x)  2  0
2 cos(5 x)  2  0
2 cos(5 x)   2
cos(5 x) 
 2
2
Now apply what we did above to get the solutions.
Solutions:
x
3 2k

20
5
and
x
5 2k

20
5
x  27

 k  72 and x  45  k  72

Simplifying Equations Before Solving
Some times equations might need to be simplified using a combination of
trigonometric identities and algebra before solving them.
Give answers in
radians.
Regroup:
Factor:
Factor:
Solve each equation
Equations with Powers of Sine or Cosine
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.
2
Solve: cos  2x   4 cos 2x   3  0
Solve: 2 sin 2 x  sin x
2 sin 2 x  sin x  0
sin x2 sin x  1  0
sin x  0 and
2 sin x  1  0
sin x  0 and
sin x 
1
2
5
6


6
0
Solutions:
cos 2x   3cos 2x   1  0
cos 2x   3  0 and cos 2x   1  0
cos 2x   3 and cos 2x   1
No Solutions
(first equation)
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
Equations With Both Sine and Cosine
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.
Solve: 5 sin 2 x  14 cos x  13  0
5(1  cos x)  14 cos x  13  0
2
36.8699
Find the other angle.
5  5 cos 2 x  14 cos x  13  0
 5 cos x  14 cos x  8  0
2
5 cos 2 x  14 cos x  8  0
(cos x  2)(5 cos x  4)  0
cos x  2  0 and
cos x  2 and
No Solution and
5 cos x  4  0
cos x 
4
5
x  cos 1  54 
x  36.8699
360-36.8699=323.13
323.13
The solutions for this are:
x  36.8699  k  360

x  323.13  k  360
