Trigonometric equation solution

1. Solving Trigonometric Equations

2. First Degree Trigonometric
Equations:
• These are equations where there is one
kind of trig function in the equation and
that function is raised to the first power.
1
)
sin(
2 

x

3. Steps for Solving:
• Isolate the Trigonometric function.
• Use exact values to solve and put answers in terms
of radians.
• If the answer is not an exact value, then use inverse
functions on your calculator to get answers

4. 1
)
sin(
2 

x
1
sin( )
2
x


Now figure out where sin = -1/2 on the unit circle.
1 7 11
sin
2 6 6
at and
 



5. Complete the List of
Solutions:
• If you are not restricted to a specific
interval and are asked to give the general
solutions then remember that adding on
any integer multiple of 2π represents a co-
terminal angle with the equivalent
trigonometric ratio.

6. 













k
k
x
Solutions
2
6
11
2
6
7
:
Where k is an integer and gives all
the coterminal angles of the
solution.

7. Practice
• Solve the equation. Find the general solutions
3 csc  2  0
3 csc  2
csc 
2
3
2
2 , 2
3 3
k k
 
  
  
3
sin
2
whichmeansthat 

8. Second Degree Trigonometric
Equations:
• These are equations that have one kind of
Trigonometric function that is squared in
the problem.
• We treat these like quadratic equations
and attempt to factor or we can use the
quadratic formula.

9. 2
: 4sin ( ) 1 0 int [0,2 )
Solve x over the erval 
 
(2sin 1)(2sin 1) 0
x x
  
This is a difference of squares and can factor
Solve each factor and you should end up with 4 solutions
1 1
sin sin
2 2
x and x

 
5 7 11
, , ,
6 6 6 6
x
   


10. Practice
2
tan 2tan 1
x x
  
2
tan 2tan 1 0
x x
  
(tan 1)(tan 1) 0
x x
  
Find the general solutions for
tan 1
x  
3 7
,
4 4
x k k
 
 
  

11. Writing in terms of 1 trig fnc
• If there is more than one trig function
involved in the problem, then use your
identities.
• Replace one of the trig functions with an
identity so there is only one trig function
being used

12. Solve the following
2
2cos sin 1 0
x x
  
2
2
2
2
2(1 sin ) sin 1 0
2 2sin sin 1 0
2sin sin 1 0
2sin sin 1 0
x x
x x
x x
x x
   
   
   
  
Replace cos2 with 1-sin2
(2sin 1)(sin 1) 0
1
sin sin 1
2
7 11
2 , 2 , 2
6 6 2
x x
x and x
x k x k x k
  
  
  

 
     

13. Solving for Multiple Angles
• Multiple angle problems will now have a
coefficient on the x, such as sin2x=1
• Solve the same way as previous problems, but
divide answers by the coefficient
• For general solutions divide 2 by the coefficient
for sin and cos. Divide by the coefficient for
tan and cot.



14. Find the general solutions for
sin3 1
x  
3
3
2
x


3
2
2
3 3
k
x


 
sin 3x +2= 1
2
2 3
x k


 

15. Practice
2cos4 3 0
Solve x  
3
cos4
2
x


5 7
4 4
6 6
x and x
 
 
5 7
,
24 2 24 2
k k
x x
   
   