Transcript Slide 1

7-6 Solving Trigonometric
Equations
Finding what x equals
It is just like solving regular equations, but
once you get solutions, you have to find
corresponding angle measure.
Lets try a problem then see the rules.
1. 2sin x  1
2
What are the rules?
1. “x” means give the answer in __________; “Θ”
means give the answer in ___________.
2. “Solve for 0 ≤ x < 2π” means give all the
answers on one pass around the unit circle.
3. “General solution” means ______________
__________________________________
4. Guess what: Work on both sides of the
equation using all the rules of algebra.
That is a) _____________ or b) ___________
Is that it?
Well, yes, except for one footnote. Never
Never Never Never divide both sides by
the same trig function to get rid of it.
For example,
sin x  sin x
2
It will eliminate answers.
Lets Try a few
2. cos2x  sin2 x  1
3. tan x  sec x  0
4. cos 2x  cos x
5. 2sin 3θ  1  0
6. sinx+cos x  0
7-6 Solving Trigonometric
Equations
Day 2
Lets go back to the solutions from yesterday
and turn them into general solutions.
• General Solutions will help you find every
single solution no matter how many times
around the circle
• All we do is add  k 2 after the answers
for one time around the circle.
• Or  k
for tangent answers because
___________________________________
___________________________________
General Solution
2. cos2x  sin2 x  1
6. tan2 x  3  0
6-5 Inverse Trig
An Inverse Function
What was that again?
Lets remember:
What is an inverse function?
What is the notation?
___________________________
In a way, you have been practicing the
inverse trig process. In section 7-6, you
had the trig value and found the angle.
f(x) = sin x
f(x) = sin-1x
x
0
y
0
x
0
y
0

1
1

2

3
2
2
0
1
0
2
0

1 3
2
0
2
What is the problem here?
______________________________
How do we take care of that?
Therefore, there are limits on the answers
that you can get.
Use your calculator to find cos-1(-.5)
_____________________________
_____________________________
Each function has a limited Range
sin-1x,
csc-1x,
tan-1x


 y
2
2
For
____________________________
0 y 
For cos-1x, sec-1x, cot-1x
____________________________
REMEMBER
With inverse trig you give only ___________
___________________________________
___________________________________
An answer in quadrant 4 such as 300 must
be given as -60. BE Careful!!
A Hint
To give yourself something to remember,
use the phrase “What angle has a” for the
symbol -1. SO, lets try some problems.
1. sin
1
2
2
 1
2. cos   
 2
1
3. arc tan 1
4. cot
1
 1
1
5. csc 2

1 3 
6. sin  tan

4

Inverse Rule
1
sin(sin
)
2
1
1

tan (tan ) 
3
6-5 Day 2
Inverse Trig
Continued
We will now combine Inverse Trig with:
•Addition and Subtraction Formulas
___________________
•Double Angle Formulas
________________
•Half Angle Formulas
___________________
8
3
cos(arcsin  arctan )
17
4
Example
A
B
tan(arctan 3  arctan 4)
5
sin(2 arctan )
2
 1 1 3 
cos  csc  
2
2