# Trigonometry : Solve a Trigonometric Function by Squaring Both Sides

## Example Questions

### Example Question #51 : Trigonometric Functions

True or False: All solutions found from squaring both sides of a trigonometric function are valid should be given as a final answer.

True

False

False

Explanation:

This is not true.  This is because when squaring both sides and then plugging back into the original equations, some of our solutions may be extraneous solutions.  Therefore, when solving a trigonometric equation by squaring both sides, all solutions found must be plugged back into the original equation and validated.

### Example Question #52 : Trigonometric Functions

True or False: You should always solve for a trigonometric equation by squaring both sides.  This will always be the most efficient method.

True

False

False

Explanation:

We should only square by both sides when all other identities are not able to be used in an equation.  Quite often, you will find that a trigonometric identity can be used to simplify an equation.  Squaring both sides is ultimately trying to produce a trigonometric identity in order to solve for the equation.

### Example Question #3 : Solve A Trigonometric Function By Squaring Both Sides

Solve the following equation by squaring both sides:

Explanation:

We begin with our original equation:

(Pythagorean Identity)

Looking at the unit circle we see that  at  and .  We must plug these back into our original equation to validate them.

Checking

Checking

And so our only solution is

### Example Question #4 : Solve A Trigonometric Function By Squaring Both Sides

Solve the following equation by squaring both sides:

Explanation:

We begin with our original equation:

(Pythagorean Identity)

From the unit circle, we see that We must check both of these solutions in the original equation.

Checking

Checking

So we see our only solution is

### Example Question #5 : Solve A Trigonometric Function By Squaring Both Sides

Solve the following equation by squaring both sides:

Explanation:

We begin with our original equation

(Pythagorean Identity)

(substitution)

Using this form, we see we really only need to consider when   at   and .  Now we must plug these values into the original equation to check and see if they are both acceptable solutions to our problem.

Checking :

Checking

By checking our solutions we see the only solution to our equation is .

### Example Question #6 : Solve A Trigonometric Function By Squaring Both Sides

Solve the following equation by squaring both sides:

Explanation:

We begin with our original equation.

(Pythagorean Identity)

(Double-Angle Formula)

We know that  will be equal to  for when  is any multiple of and when .  We need to check both solutions (we will simply check  for simplicity) to make sure they are valid solutions.

Checking :

Checking

By checking our solutions, we see the only solution to this equation is

### Example Question #7 : Solve A Trigonometric Function By Squaring Both Sides

Solve the following equation by squaring both sides: .

Explanation:

This one is not as straight-forward.  We must manipulate the original equation before squaring both sides.

(Pythagorean Identity)

(divide both sides by 2)

Solving for each:

Or

From the unit circle we know that  when .

So now we must go back and check all of our solutions.

Checking

Checking  (this is also equal to checking )

Both of our solutions are correct.

### Example Question #61 : Trigonometric Functions

Which of the following is the main purpose of squaring both sides of a trigonometric equation?

To solve the problem, duh!

To produce a familiar identity/formula that we can use to solve the problem

To get rid of a radical

Working with square trigonometric functions is easier than those of the first power