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