Some equations have multiple terms with trigonometric functions. In that case, you have to use the trigonometric identities to group the ’s. Here are some examples of using trigonometric identities in equations.
Example 1
Equations in the Form
In this case, you divide by on both sides of the equation to get an expression with . This works because . Lastly, in this example, you have to check the case where to make sure you don’t miss any solutions.
You are solving the equation
(1) |
for .
In the interval , that gives you the solutions
Now you have to test these answers in the main equation (1):
:
:
:
:
:
:
:
:
Equation (1) doesn’t have any more solutions, and you can conclude that the solutions in the interval are:
Example 2
Equations in the form
require you to use the identities
and
Solve the equation
You use the zero product property and look at the two equations and . The equation has the solutions
You rewrite the equation as the basic equation . It has the solutions
That means the solutions for the interval are
Example 3
Equations in the form
require you to use the identity
and substitution.
Solve the equation
You make the substitution in this equation, which gives you
Solve the quadratic equation:
Now you substitute back in for :
Note that is not included among the solutions. That’s because it’s outside the interval we’re looking at.