Download 6.3

Solving Trigonometric Equations
Involving Multiple Angles
6.3
JMerrill, 2009
Strategies for Solving Trig.
Equations with Multiple Angles
• If the equation involves functions of 2x
and x, transform the functions of 2x into
functions of x by using identities
• If the equation involves functions of 2x
only, it is usually better to solve for 2x
directly and then solve for x
• Be careful not to lose roots by dividing off
a common factor
• Remember: You can always graph to check
your solutions
Example
• Solve cos 2x = 1 – sin x for 0 ≤ x < 2π
cos2x  1  sin x
1  2 sin2 x  1  sin x
1  2 sin2 x  1  sin x  0
sin x (2 sin x  1)  0
2 sin2 x  sin x  0
sin x  0
x  0, 
1
sin x 
2
 5
x  ,
6 6
You Do
• Solve for 0o≤θ<360o
cos 2x = cos x
2cos2 x  1  cos x  0
2cos2 x  cos x  1  0
(2cos x  1)(cos x  1)  0
1
cos x  ,cos x  1
2
120o , 240o , 0o
Example
• Solve 3cos2x + cos x = 2 for 0 ≤ x < 2π
3cos2x  cos x  2
3(2cos2 x  1)  cos x  2
6cos2 x  3  cos x  2
6cos x  cos x  5  0
2
(6cos x  5)(cos x  1)  0
5
cos x 
cos x  1
6
x  0.59, 5.70 x    3.14
All of the previous examples were solved
for x. Now we’ll solve for 2x directly.
Example
• Solve 2sin2x = 1 for 0o ≤ θ < 360o
2 sin 2x  1
Pretend the 2 isn’t in
front of the x and solve
it (solve sin x = ½ )
1
sin 2x 
2
2x  30 ,150 ,390 ,510
o
o
o
x  15o , 75o ,195o ,255o
0
You Do
• Solve for 0o≤θ<360o
tan22x-1=0
tan 2x  1
2
tan2x  1
2x  45o ,135o ,225o ,315o ,
405o , 495o ,585o ,675o ,
x  22.5o ,67.5o ,112.5o ,157.5o ,
202.5o ,247.5o ,292.5o ,337.5o ,