# Trigonometry : Solving Trigonometric Equations

## Example Questions

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### Example Question #1 : Systems Of Trigonometric Equations

For this question, we will denote by max the maximum value of the function and min the minimum value of the function.

What is the maximum and minimum values of

where  is a real number.

Explanation:

To find the maximum and the minimum , we can view the above function as

a system where and . Using these two conditions we find the maximum and the minimum.

means also that () We also have:

implies that :

() Therefore we have by adding () and()

This means that max=2 and min=-1

### Example Question #2 : Systems Of Trigonometric Equations

Find the values of  that satisfy the following system:

where  is assumed to be

This system does not have a solution.

This system does not have a solution.

Explanation:

We can write the system in the equivalent form:

The solution to the first equation is

means that

This means that there is no x that satisfies the system.

Therefore there is no x that solves the 3 inequalities simultaneously.

### Example Question #1 : Systems Of Trigonometric Equations

Which of the following systems of trigonometric equations have a solution with an -coordinate of ?

More than one of these answers has a solutions at .

Explanation:

The solution to the correct answer would be .

For all of the other answers, plugging in for the second equation gives a y value of .

### Example Question #4 : Systems Of Trigonometric Equations

Solve the system for :

no solution

Explanation:

First, set both equations equal to each other:

subtract from both sides

Now we can solve this as a quadratic equation, where "x" is . Using the quadratic formula:

This gives us 2 potential solutions for :

the sine of an angle cannot be greater than 1

### Example Question #1 : Solving Trigonometric Equations

Solve this system for :

Explanation:

First, set the two equations equal to each other

subtract the sine term from the right

subtract 3 from both sides

divide by 2

multiply by 2

### Example Question #6 : Systems Of Trigonometric Equations

Solve this system for :

no solution

Explanation:

Set the two equations equal to each other

subtract cos from both sides

take the square root of both sides

### Example Question #7 : Systems Of Trigonometric Equations

Solve this system for :

Explanation:

Set both equations equal to each other:

subtract from both sides

subtract from both sides

We can re-write the left side using a trigonometric identity

take the inverse cosine

divide by 2

### Example Question #8 : Systems Of Trigonometric Equations

Solve this system for :

no solution

Explanation:

Set the two equations equal to each other:

subtract from both sides

divide both sides by 4

take the square root of both sides

### Example Question #9 : Systems Of Trigonometric Equations

Solve the following system:

The system does not have a solution.

The system does not have a solution.

Explanation:

A number x is a solution if it satisfies both equations.

We note first we can write the first equation in the form :

We know that for all reals. This means that there is no x that

satisifies the first inequality. This shows that the system cannot satisfy both equations since it does not satisfy one of them. This shows that our system does not have a solution.

### Example Question #10 : Systems Of Trigonometric Equations

Solve this system for :

Explanation:

First, set both equations equal to each other:

subtract from both sides

Using a trigonometric identity, we can re-write as :

combine like terms

subtract 2 from both sides

We can solve for using the quadratic formula:

This gives us 2 possible values for cosine

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