Transcript 5 3 Solving Trig Eqs II

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 2x 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.
x3  x
Solve for x.
cos3 x  cos x
Solve for x.
x 3  x 2  3x  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  3x  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  2  sin x
Solve for x.
cos x  sin x tan 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 sin2 x, which can be replaced
with 1 – cos2 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:
2
tan x 
3
2
x  arctan  
3
Now find the coterminal angles:
2
x  arctan    n
3
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 3x  1
2 sin k  1 , where k = 3x
1
Now solve for k: sin k 
2

k   2 n
6
2 sin 3x  1
2 sin k  1 , where k = 3x
1
Of course,
Now solve for k: sin k 
2
you’d have to
do this for the

k   2 n
other set of
6
solutions, too.

Finally “unsubstitute”
Just saying.
3 x   2 n
k and solve for x:
6
 2 n
x 
18
3
Solve for x.
3
sin 2 x 
0
2
Solve for x.
 x
tan    1  0
2
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