1. Solving Trigonometric Equations

2. Solving Trigonometric Equations
For most problems,
The solution interval
Will be
[0, 2)
You are responsible for checking your solutions back into the original problem!

3. 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.

4. Now figure out where sin = -1/2 on the unit circle.

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. Where k is an integer and gives all the coterminal angles of the solution.

7. Practice
Solve the equation. Find the general solutions

8. Solving Trigonometric Equations
Solve:
Step 1: Isosolate cos x using algebraic skills.
Step 2: Determine in which quadrants cosine is positive. Use the inverse
function to assist by finding the angle in Quad I first. Then use that angle
as the reference angle for the other quadrant(s). QI
QIV
Note: cosine is positive in
Quad I and Quad IV.
Note: The reference angle is /3.

9. Solving Trigonometric Equations
Solve:
Step 1:
Note: Since there is a  , all four quadrants
hold a solution with /4 being the reference
angle.
Step 2: Q1
QII
QIII
QIV

10. Solving Trigonometric Equations
Solve:
Step 1:
Step 2:
Note: There is no solution here because 2
lies outside the range for cosine.

11. Solving Trigonometric Equations
Try these:
Solution 1. 2. 3.

12. Solving Trigonometric Equations
Solve:
Factor the quadratic equation.
Set each factor equal to zero.
Solve for sin x
Determine the correct quadrants
for the solution(s).

13. This is a difference of squares and can factor
Solve each factor and you should end up with 4 solutions

14. Practice
Find the general solutions for

15. 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

16. Solve the following
Replace cos2 with 1 - sin2

17. Solving Trigonometric Equations
Solve:
Replace sin2x with 1-cos2x
Distribute
Combine like terms.
Multiply through by – 1.
Factor.
Set each factor equal to zero.
Solve for cos x.
Determine the solution(s).

18. Solving Trigonometric Equations
Solve:
Square both sides of the equation
in order to change sine into terms
of cosine giving only one trig
function to work with.
FOIL or Double Distribute
Replace sin2x with 1 – cos2x
Set equation equal to zero since it is a
quadratic equation.
Factor
Set each factor equal to zero.
Solve for cos x X
Determine the solution(s).
It is removed because it does not
check in the original equation.
Why is 3/2 removed as a solution?

19. Solving Trigonometric Equations
Solve:
No algebraic work needs to be done because cosine is already by itself.
Remember, 3x refers to an angle and one cannot divide by 3 because it
is cos 3x which equals ½.
Solution:
Since 3x refers to an angle, find the angles whose cosine value is ½.
Now divide by 3 because it is angle equaling angle.
Notice the solutions do not exceed 2. Therefore,
more solutions may exist.
Return to the step where you have 3x equaling
the two angles and find coterminal angles for
those two.
Divide those two new angles by 3.

20. Solving Trigonometric Equations
The solutions still do not exceed 2.
Return to 3x and find two more
coterminal angles.
Divide those two new angles by 3.
The solutions still do not exceed 2.
Return to 3x and find two more
coterminal angles.
Divide those two new angles by 3.
Notice that 19/9 now exceeds 2 and
is not part of the solution.
Therefore the solution to cos 3x = ½ is

21. Solving Trigonometric Equations
Try these:
Solution 1. 2. 3. 4.

22. Find the general solutions for
sin 3x +2= 1

23. Practice