Solving Trigonometric Equations Involving Multiple Angles 6.3 JMerrill, 2009

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