Trigonometry : Systems of Trigonometric Equations

Study concepts, example questions & explanations for Trigonometry

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Example Questions

Example Question #1 : Systems Of Trigonometric Equations

 

Solve the following sytem:

Possible Answers:

The system does not have a solution.

Correct answer:

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 #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.

Possible Answers:

Correct answer:

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

Possible Answers:

This system does not have a solution.

Correct answer:

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 #3 : Systems Of Trigonometric Equations

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

Possible Answers:

More than one of these answers has a solutions at .

Correct answer:

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 :

Possible Answers:

no solution

Correct answer:

Explanation:

First, set both equations equal to each other:

 subtract from both sides

add 1 to 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 #5 : Systems Of Trigonometric Equations

Solve this system for :

Possible Answers:

Correct answer:

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

 

Example Question #31 : Trigonometric Equations

Solve this system for :

Possible Answers:

Correct answer:

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 #32 : Trigonometric Equations

Solve this system for :

Possible Answers:

no solution

Correct answer:

Explanation:

Set the two equations equal to each other

subtract cos from both sides

take the square root of both sides

Example Question #8 : Systems Of Trigonometric Equations

Solve this system for :

Possible Answers:

Correct answer:

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 #9 : Systems Of Trigonometric Equations

Solve this system for :

Possible Answers:

no solution

Correct answer:

Explanation:

Set the two equations equal to each other:

subtract from both sides

add 5 to both sides

divide both sides by 4

take the square root of both sides

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