This document discusses different types and methods for solving trigonometric equations. It covers first degree equations with one trig function raised to the first power, second degree equations with one trig function squared, and equations with multiple trig functions that require using identities. The steps involve isolating the trig function, using inverse functions to find solutions, factoring when possible, and finding the general solutions by adding integer multiples of 2π or appropriate coefficients. Practice problems are provided with worked out solutions.

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
   
   