Chapter 7Section 7.6

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

2)5cos(

x

22)5cos( x

43

45

1. Draw unit circle.

2. Draw horizontal or vertical line the correct distance on x or y axis.

3. Find angles where line hits the unit circle.

4. Add 2k to each angle.

5. Solve for x.

22

Solutions:

3602255360135524

552

4

35 kxandkxkxandkx

22.. cei

22

724572275

2

20

5

5

2

20

3 kxandkx

kxand

kx

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:

02)5cos(2 x22)5cos(

2)5cos(2

02)5cos(2

x

x

xNow apply what we did above to get the solutions.

Solutions:

724572275

2

20

5

5

2

20

3 kxandkx

kxand

kx

Simplifying Equations Before Solving

Sometimes equations might need to be simplified using a combination of trigonometric identities and algebra before solving them.

Regroup:

Factor:

Factor:

Solve each equation

Give answers in radians.

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.

Solve: xx sinsin2 2

21

2

sin0sin

01sin20sin

01sin2sin

0sinsin2

xandx

xandx

xx

xx

06

65

Solutions:

kxandkx

kxandkx

22

220

65

6

Solve: 03cos4cos 222 xx

1cos3cos

01cos03cos

01cos3cos

22

22

22

xx

xx

xx

and

and

Solutions:

kx

kx

42

22

No Solutions(first equation)

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: 013cos14sin5 2 xx

8699.36

cos

cos2cos

04cos502cos

0)4cos5)(2(cos

08cos14cos5

08cos14cos5

013cos14cos55

013cos14)cos1(5

541

54

2

2

2

2

x

xandSolutionNo

xandx

xandx

xx

xx

xx

xx

xx

8699.36

Find the other angle.

360-36.8699=323.13

13.323

The solutions for this are:

36013.323

3608699.36

kx

kx