SolvingSolving Trigonometric Trigonometric EquationsEquations
Involving Multiple AnglesInvolving Multiple Angles6.36.3
SolvingSolving Trigonometric Trigonometric EquationsEquations
Involving Multiple AnglesInvolving Multiple Angles6.36.3
JMerrill, 2009JMerrill, 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
< 2π2
2
2
cos2 1 sin
1 2sin 1 sin
1 2sin 1 sin 0
2sin sin 0
x x
x x
x x
x x
sin (2sin 1) 0
1sin 0 s
50,
i2
6
n
,6
x
x x
x x
x
You Do• Solve for 0o≤θ<360o
cos 2x = cos x2
2
cos 0
2cos cos 1 0
(2cos 1)(cos 1) 0
1cos ,cos 1
2120 ,
2c
240
s
0
o 1
,o o o
x
x x
x x
x x
x
Example• Solve 3cos2x + cos x = 2 for 0 ≤ x
< 2π2
2
2
3cos2 cos 2
3(2cos 1) cos 2
6cos 3 cos 2
6cos cos 5 0
x x
x x
x x
x x(6cos 5)(cos 1) 0
5cos
0.5
cos 169,5.70 3.14
x x
x x
x x
Example• Solve 2sin2x = 1 for 0o ≤ θ < 360o
2sin2 1
1sin2
2
x
x
Pretend the 2 isn’t in front of the x and solve it (solve sin x = ½ )
02 30 ,150 ,390 ,51
15 ,75 ,195 ,255
0
o o o o
o o o
x
x
All of the previous examples were solved for x. Now we’ll solve for 2x directly.
You Do
• Solve for 0o≤θ<360o
tan22x-1=02tan 2 1
tan2 1
2 45 ,135 ,225 ,315 ,
405 ,495 ,585 ,675 ,
22.5 ,67.5 ,112.5 ,157.5 ,
202.5 ,247.5 ,292.5 ,337.5 ,
o o o o
o o o o
o o o o
o o o o
x
x
x
x