When solving trigonometric equations, it’s very important to include all the solutions! That means you have to be aware of the period of the function you are working with, and how that influences the number of solutions.
Rule
When and , the following is true:
Note! It’s very important to check which values is allowed to have. It varies from problem to problem, and influences which values for you can use in your answer.
Example 1
Solve the equation for
You begin by transforming the equation to get the -term on its own:
This has the solutions
First, you continue with (1):
Then you continue with (2):
The problem tells you to find all the solutions that are in the interval . You find these by considering and with respect to that interval.
Look at first. If you insert , you get
which is in the interval. When you check , you get
Then you check ,
which is also in the interval. Now you notice that if you check , the answer will be outside the interval . That means you have found all the solutions for .
Now you have to do the same for . The values still have to be in the interval for them to be a part of the solution. That gives you
As you can see, the last value is outside the interval. That means the solutions in the interval are:
Example 2
Solve the equation for
The basic equation
has the solutions
First, you continue working on (3):
Then you work on (4):
Now you have to find the solutions from and . The values have to be in the interval for them to be one of the solutions. For , you have
where is outside the interval. When you check , you get
where is outside the interval. That means solutions in the interval are:
Example 3
Solve the equation for
You solve the trigonometric equation for :