3.4: Trigonometric Equations Using Factoring
Solving trig equations is an important process in mathematics. Quite often you'll see powers of trigonometric functions and be asked to solve for the values of the variable which make the equation true. For example, suppose you were given the trig equation
Could you solve this equation? (You might be tempted to just divide both sides by
Watch This
James Sousa Example: Solve a Trig Equation by Factoring
Guidance
You have no doubt had experience with factoring. You have probably factored equations when looking for the possible values of some variable, such as "x". It might interest you to find out that you can use the same factoring method for more than just a variable that is a number. You can factor trigonometric equations to find the possible values the function can take to satisfy an equation.
Algebraic skills like factoring and substitution that are used to solve various equations are very useful when solving trigonometric equations. As with algebraic expressions, one must be careful to avoid dividing by zero during these maneuvers.
Example A
Solve
Solution:
Example B
Solve
Solution:
Pull out
There is a common factor of
Think of the
Example C
Solve
Solution:
Some trigonometric equations have no solutions. This means that there is no replacement for the variable that will result in a true expression.
Vocabulary
Factoring: Factoring is a way to solve trigonometric equations by separating the equation into two terms which, when multiplied together, give the original expression. Since the product of the two factors is equal to zero, each of the factors can be equal to zero to make the original expression true. This leads to solutions for the original expression.
Guided Practice
1. Solve the trigonometric equation
2. Solve
3. Find all the solutions for the trigonometric equation
Solutions:
1. Use factoring by grouping.
2.
3.
Concept Problem Solution
The equation you were given is
To solve this:
Subtract
Now set each factor equal to zero and solve. The first is
And now for the other term:
Practice
Solve each equation for

cos2(x)+2cos(x)+1=0 
1−2sin(x)+sin2(x)=0 
2cos(x)sin(x)−cos(x)=0 
sin(x)tan2(x)−sin(x)=0 
sec2(x)=4 
sin2(x)−2sin(x)=0 
3sin(x)=2cos2(x) 
2sin2(x)+3sin(x)=2 
tan(x)sin2(x)=tan(x) 
2sin2(x)+sin(x)=1  \begin{align*}2\cos(x)\tan(x)\tan(x)=0\end{align*}
 \begin{align*}\sin^2(x)+\sin(x)=2\end{align*}
 \begin{align*}\tan(x)(2\cos^2(x)+3\cos(x)2)=0\end{align*}
 \begin{align*}\sin^2(x)+1=2\sin(x)\end{align*}
 \begin{align*}2\cos^2(x)3\cos(x)=2\end{align*}
Notes/Highlights Having trouble? Report an issue.
Color  Highlighted Text  Notes  

Please Sign In to create your own Highlights / Notes  
Show More 
Image Attributions
Here you'll learn how to factor trig equations and then solve them using the factored form.