Objectives:
1.
To solve trig equations
Assignment:
• P. 396: 21-34 S
• P. 396: 35-40 S
• P. 369: 42
• P. 369: 59-62 S
• P. 370: 74
• Project Sneak Peek:
Another Special Right
Triangle

You will be able to solve trig equations

Remember that when you solve a trig equation, your goal is to isolate the trig expression.
• Sometimes this will require factoring if there is more than one trig expression
• Here, you want to get a product of factors to equal zero and apply the Zero Product
Property, just like solving polynomial equations

Once your trig expression is isolated, now it’s time to work backwards to find the angle measure that yields the correct ratio.
• You could take the inverse, but that would only give you one answer
• To find more answers, find the coterminal angles

Once your trig expression is isolated, now it’s time to work backwards to find the angle measure that yields the correct ratio.
• To find even more answers, use the unit circle to find another set of angles that give the exact same ratio

Here are some things you may have to try while solving trig equations:
1.
Getting all expressions on one side of the equation and factoring
2.
Converting all trig expressions to the same expression (all tangent, for example)
3.
Converting all trig expressions to sines and cosines

4.
Getting a common denominator
5.
Squaring both sides
6.
Use inverses (arctan, arcsin, arccos) for those values that we don’t know from the unit circle
7.
Dealing with multiple angles (sin 2 x instead of sin x )

When factoring, you first want to get everything on one side of the equation. Then use your factoring skills to:
• Factor out a GCF
• Reverse FOIL, complete the square, use the quadratic formula
• Factor by grouping
• Use the Rational Zero Test for higher order polynomials

Solve for x .
x
3  x

Solve for x .
cos
3 x
 cos x

Solve for x .
x
3  x
2 
3 x

3

Solve for x .
tan
3 x
 tan
2 x

3 tan x

3

Solve for x .
8 x
4 
4 x
3 
10 x
2 
3 x

3

0

Solve for x .
8 sin
4 x

4 sin
3 x

10 sin
2 x

3 sin x

3

0

Sometimes you have to simplify your trig expressions before solving the equation:
• Convert to the same trig function (all tangents)
• Convert to all sines and cosines
• Use Fundamental Identities
When you substitute one trig function for another, sometimes you’ll get something that doesn’t actually work. So throw it out.

Solve for x .
2 cos
2 x x

Solve for x .
cos x
 x x

2

You might come across an equation like the one below that you can’t directly simplify.
sin x
 cos x
Try squaring both sides of the equation:
• Turns sin x into sin 2 x , which can be replaced with 1 – cos 2 x
• If you have to square both sides, you can introduce extraneous solutions that you have to check in the original equation

Solve for x .
sin x
 cos x

Solve for x .
sec x
 tan x

1

So far we’ve only solved trig equations whose solutions were conveniently located on the unit circle. When you can’t easily find your answers on the unit circle, just use inverses: tan x
 x

2
3 arctan



2
3 


Now find the coterminal angles: x
 arctan
2
3
 n

Of course, you could use a calculator for the value of arctan (2/3)

Solve for x .
tan
2 x

5 tan x

6

Solve for x .
sec
2 x

2 tan x

4

If your equation is in terms of a multiple of x , just do a substitution with your favorite variable and then deal with it at the end.
2 sin 3 x

1
Now solve for k :
2 sin k

1 , where k = 3 x sin k

1
2
 k

6

2 n


2 sin 3 x

1 2 sin k

1 , where k = 3 x
Now solve for k :
Finally “unsubstitute” k and solve for x : sin k
 k
3 x x
1



2


2 n

6



2 n
6

2 n


18 3
Of course, you’d have to do this for the other set of solutions, too.
Just saying.

Solve for x .
sin 2 x

2
3

0

Solve for x .
tan


 x
2 



1

0

Objectives:
1.
To solve trig equations
Assignment:
• P. 396: 21-34 S
• P. 396: 35-40 S
• P. 369: 42
• P. 369: 59-62 S
• P. 370: 74
• Project Sneak Peek:
Another Special Right
Triangle