Call Now to Set Up Tutoring:

(888) 888-0446

Trigonometry : Solving Trigonometric Equations

Study concepts, example questions & explanations for Trigonometry

Create An Account

All Trigonometry Resources

6 Diagnostic Tests 155 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #185 : Trigonometry

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

cos^4x+sin^3x, where is a real number.

Possible Answers:

max=0, min=-1

max=1, min=0

max=2, min=-2

max=2, min=-1

max=1, min=-1

Correct answer:

max=2, min=-1

Explanation:

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

a system where $-1\le cosx\le 1$ and $-1\le sinx\le 1$ . Using these two conditions we find the maximum and the minimum.

$0\le cos^2x\le 1$ means also that $0\le cos^4x\le 1$ ( $\star$ ) We also have:

$-1\le sinx\le 1$ implies that :

$-1\le sin^{3}x\le 1$ ( $\star \star$ ) Therefore we have by adding ( $\star$ ) and( $\star \star$ )

$-1\le cos^4x+sin^3x\le 2$

This means that max=2 and min=-1

Report an Error

Example Question #1 : Systems Of Trigonometric Equations

Find the values of that satisfy the following system:

$\left\{\begin{matrix} 2cosx\le \sqrt{2}\\2sinx\le \sqrt{2} \\ cosx< sinx\end{matrix}\right.$

where is assumed to be $[0,\frac{\pi}{2}]$

Possible Answers:

$[-\frac{\pi}{4},\frac{\pi}{2}]$

$[0,\frac{\pi}{4}]$

$[0,\frac{\pi}{2}]$

This system does not have a solution.

$[\frac{\pi}{4},\frac{\pi}{2}]$

Correct answer:

This system does not have a solution.

Explanation:

We can write the system in the equivalent form:

$cosx\le \frac{\sqrt{2}}{2}$

$sinx\le \frac{\sqrt{2}}{2}$

cosx< sinx

The solution to the first equation is $x\in \[0,\frac{\pi}{4}]$

cosx< sinx means that $x\in (\frac{\pi}{4},\frac{\pi}{2}]$

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

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

Report an Error

Example Question #191 : Trigonometry

Which of the following systems of trigonometric equations have a solution with an -coordinate of $\frac{\pi}{4}$ ?

Possible Answers:

y=sin(x)

y=cos(3x)

y=sin(x)

y=cos(5x)

y=sin(x)

y=cos(7x)

More than one of these answers has a solutions at $x=\frac{\pi}{4}$ .

Correct answer:

y=sin(x)

y=cos(7x)

Explanation:

The solution to the correct answer would be $\left(\frac{\pi}{4}, \frac{1}{\sqrt2}\right)$ .

For all of the other answers, plugging in $\frac{\pi}{4}$ for the second equation gives a y value of $-\frac{1}{\sqrt2}$ .

Report an Error

Example Question #2 : Solving Trigonometric Equations

Solve the system for $\theta$ :

$\begin{matrix} y = 6 \sin \theta - 1 \\ y = \sin ^2 \theta + 4 \end{matrix}$

Possible Answers:

no solution

$\frac{3 \pi }{2}$

$\pi$

$\frac{ \pi }{2 }$

$2 \pi$

Correct answer:

$\frac{ \pi }{2 }$

Explanation:

First, set both equations equal to each other:

$6 \sin \theta - 1 = \sin ^2 \theta + 4$ subtract $6 \sin \theta$ from both sides

$-1 = \sin ^2 \theta - 6 \sin \theta + 4$ add 1 to both sides

$0 = \sin ^2 \theta - 6 \sin \theta + 5$

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

$\sin \theta = \frac{ 6 \pm \sqrt{ 36 - 4(1)(5)}}{2} = \frac{6 \pm \sqrt{16 }}{2} = \frac{ 6 \pm 4 }{2}$

This gives us 2 potential solutions for $\sin \theta$ :

$\sin \theta = \frac{10}{2} = 5$ the sine of an angle cannot be greater than 1

$\sin \theta = \frac{2}{2 } = 1$

$\theta = \sin ^ {-1} (1) = \frac{ \pi }{2}$

Report an Error

Example Question #3 : Solving Trigonometric Equations

Solve this system for $\theta$ :

$\begin{matrix} y = 5 \sin (\frac{ \theta }{2}) + 3 \\ y = 3 \sin (\frac{\theta }{2} ) + 2 \end{matrix}$

Possible Answers:

$\frac{ 11 \pi }{12} , \frac{ 7 \pi }{12}$

$\frac{ \pi }{3} , \frac{ 5 \pi }{3}$

$\frac{ \pi }{6} , \frac{ 5 \pi }{6}$

$\frac{ 11 \pi }{6} , \frac{ 7 \pi }{6}$

$\frac{ 11 \pi }{3} , \frac{ 7 \pi }{3}$

Correct answer:

$\frac{ 11 \pi }{3} , \frac{ 7 \pi }{3}$

Explanation:

First, set the two equations equal to each other

$5 \sin (\frac{ \theta }{2}) + 3 = 3 \sin (\frac{ \theta }{2} ) + 2$ subtract the sine term from the right

$2 \sin (\frac{ \theta } { 2 }) + 3 = 2$ subtract 3 from both sides

$2 \sin (\frac{ \theta }{2} ) = -1$ divide by 2

$\sin (\frac{ \theta }{2} ) = -\frac{1}{2}$

$\frac{ \theta } {2} = \sin ^{-1} (- \frac{1}{2}) = \frac{ 11 \pi }{6} , \frac{ 7 \pi }{6}$ multiply by 2

$\theta = \frac{ 11 \pi }{3} , \frac{ 7 \pi }{3}$

Report an Error

Example Question #1 : Solving Trigonometric Equations

Solve this system for $\theta$ :

$y = \sin ^ 2 \theta + \cos \theta$

$y = \cos \theta + \frac{1}{2}$

Possible Answers:

$\frac{ \pi }{4} , \frac{ 3 \pi }{4} ,\frac{ 5 \pi }{4} , \frac{ 7 \pi }{4 }$

$\frac{ 3 \pi }{4} , \frac{ 5 \pi }{4}$

$\frac{ \pi }{4} , \frac{ 7 \pi }{4}$

$\frac{ \pi }{4} , \frac{ 3 \pi }{4}$

no solution

Correct answer:

$\frac{ \pi }{4} , \frac{ 3 \pi }{4} ,\frac{ 5 \pi }{4} , \frac{ 7 \pi }{4 }$

Explanation:

Set the two equations equal to each other

$\sin ^2 \theta + \cos \theta = \cos \theta + \frac{ 1}{2}$ subtract cos from both sides

$\sin ^ 2 \theta = \frac{ 1}{2 }$ take the square root of both sides

$\sin \theta = \pm \frac{ 1}{ \sqrt2}$

$\theta = \sin ^{-1} ( \pm \frac{ 1} {\sqrt2}) = \frac{ \pi }{4} , \frac{ 3 \pi }{4} , \frac{ 5 \pi }{4} , \frac{ 7 \pi }{4}$

Report an Error

Example Question #31 : Trigonometric Equations

Solve this system for $\theta$ :

$\\ y = \cos ^2 \theta + \frac{1}{2} \\ y = \sin ^2 \theta$

Possible Answers:

$\frac{ \pi }{3} , \frac{ 2 \pi }{3}$

$\frac{4 \pi }{3} , \frac{ 2 \pi }{3}$

$\frac{ 7 \pi }{6} , \frac{ 11 \pi }{6}$

$\frac{ 2 \pi }{3 } , \frac{4 \pi }{3}$

$\frac{ \pi }{6 } , \frac{ 5 \pi }{6}$

Correct answer:

$\frac{ \pi }{3} , \frac{ 2 \pi }{3}$

Explanation:

Set both equations equal to each other:

$\cos ^2 \theta + \frac{1}{2} = \sin ^2 \theta$ subtract $\frac{1}{2}$ from both sides

$\cos ^2 \theta = \sin ^2 \theta - \frac{1}{2}$ subtract $\sin ^2 \theta$ from both sides

$\cos ^2 \theta - \sin ^2 \theta = - \frac{1}{2}$

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

$\cos ( 2 \theta ) = -\frac{1}{2 }$ take the inverse cosine

$2 \theta = \cos ^ {-1}\left(-\frac{1}{2} \right)$

$2 \theta = \frac{ 2 \pi }{3} , \frac{4 \pi }{3 }$ divide by 2

$\theta = \frac{\pi }{3} , \frac{ 2 \pi }{3}$

Report an Error

Example Question #2 : Systems Of Trigonometric Equations

Solve this system for $\theta$ :

$\\y = 4 \cos ^ 2 \theta + \cos \theta - 5 \\ y = \cos \theta - 2$

Possible Answers:

no solution

$\frac{ \pi }{6} , \frac{ 11 \pi }{6}$

$\frac{ \pi }{3} , \frac{ 11 \pi }{3}$

$\frac{ \pi }{6} , \frac {5 \pi }{6} , \frac{ 7 \pi }{6} , \frac{ 11 \pi }{6}$

$\frac{ \pi }{3} , \frac{ 2 \pi }{3}$

Correct answer:

$\frac{ \pi }{6} , \frac{ 11 \pi }{6}$

Explanation:

Set the two equations equal to each other:

$4 \cos ^2 \theta + \cos \theta - 5 = \cos \theta - 2$ subtract $\cos \theta$ from both sides

$4 \cos ^ 2 \theta - 5 = -2$ add 5 to both sides

$4 \cos ^ 2 \theta = 3$ divide both sides by 4

$\cos ^ 2 \theta = \frac{ 3}{4}$ take the square root of both sides

$\cos \theta = \sqrt{\frac{ 3}{4} } = \frac{ \sqrt{3} }{\sqrt{4}} = \pm \frac{ \sqrt{3}}{2}$

$\theta = \cos ^ {-1} \left( \pm \frac{ \sqrt 3}{2}\right)$

$\theta = \frac{ \pi }{6} , \frac{ 11 \pi }{6}$

Report an Error

Example Question #1 : Solving Trigonometric Equations

Solve the following system:

Screen shot 2020 05 21 at 10.30.36 am

Possible Answers:

$\frac{-\pi}{3}$

$\pi$

The system does not have a solution.

$-\pi$

$\frac{\pi}{3}$

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 :

$cosx\ge \frac{4}{3}$

We know that $cosx\le 1$ 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.

Report an Error

Example Question #1 : Solving Trigonometric Equations

Solve this system for $\theta$ :

$\begin{matrix} y = \sin ^2 \theta + 1 \\ y = \cos \theta + 2 \end{matrix}$

Possible Answers:

$\frac{ \pi }{2} , \frac{ 3 \pi }{2 }$

$\frac{ \pi }{2} , \pi, 2 \pi$

$\pi, \frac{\pi}{2} , \frac{ 3 \pi }{2}$

$\pi, \frac{ \pi }{2}$

$\frac{ 3 \pi }{2} , 2 \pi$

Correct answer:

$\pi, \frac{\pi}{2} , \frac{ 3 \pi }{2}$

Explanation:

First, set both equations equal to each other:

$\sin ^2 \theta + 1 = \cos \theta + 2$ subtract $\cos \theta$ from both sides

$\sin ^2 \theta - \cos \theta + 1 = 2$

Using a trigonometric identity, we can re-write $\sin^2 \theta$ as $1 - \cos ^2 \theta$ :

$1 - \cos^2 \theta - \cos \theta + 1 =2$ combine like terms

$- \cos ^2 \theta - \cos \theta + 2 = 2$ subtract 2 from both sides

$- \cos^2 \theta - \cos \theta = 0$

We can solve for $\cos \theta$ using the quadratic formula:

$\cos \theta = \frac{ 1 \pm \sqrt{ 1 - 4(-1)(0)}}{2(-1)} = \frac{ 1 \pm 1}{ -2 }$

This gives us 2 possible values for cosine

$\cos \theta = \frac{ 1 + 1 }{ -2 } = \frac{ 2 }{ - 2 } = -1$

$\theta = \cos ^ {-1} ( -1 ) = \pi$

$\cos \theta = \frac{ 1 - 1 }{-2 } = \frac{ 0 }{-2} = 0$

$\theta = \cos ^ {-1} (0) = \frac{ \pi }{2} , \frac{ 3 \pi }{2}$

Report an Error

View Trigonometry Tutors

Ivory
Certified Tutor

Hopkinsville Community College, Associate, Associates in Science. Western Kentucky University, Bachelor in Arts, Psychology.

View Trigonometry Tutors

Ian
Certified Tutor

Lipscomb University, Bachelor of Science, Biology, General.

View Trigonometry Tutors

Leonard
Certified Tutor

Florida State University, Bachelor of Science, Environmental Science.

All Trigonometry Resources

6 Diagnostic Tests 155 Practice Tests Question of the Day Flashcards Learn by Concept

Trigonometry Tutors in Top Cities:

Popular Courses & Classes

ISEE Courses & Classes in Chicago, SAT Courses & Classes in Dallas Fort Worth, Spanish Courses & Classes in New York City, ISEE Courses & Classes in Philadelphia, GMAT Courses & Classes in Denver, ISEE Courses & Classes in San Francisco-Bay Area, ISEE Courses & Classes in Atlanta, Spanish Courses & Classes in Dallas Fort Worth, LSAT Courses & Classes in Washington DC, GRE Courses & Classes in San Francisco-Bay Area

Popular Test Prep

GRE Test Prep in Phoenix, ACT Test Prep in Los Angeles, ISEE Test Prep in Seattle, ISEE Test Prep in New York City, GMAT Test Prep in Chicago, MCAT Test Prep in Los Angeles, LSAT Test Prep in San Francisco-Bay Area, LSAT Test Prep in Chicago, MCAT Test Prep in San Diego, SAT Test Prep in Atlanta

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

Trigonometry : Solving Trigonometric Equations

Study concepts, example questions & explanations for Trigonometry

All Trigonometry Resources

Example Questions

Example Question #185 : Trigonometry

Example Question #1 : Systems Of Trigonometric Equations

Example Question #191 : Trigonometry

Example Question #2 : Solving Trigonometric Equations

Example Question #3 : Solving Trigonometric Equations

Example Question #1 : Solving Trigonometric Equations

Example Question #31 : Trigonometric Equations

Example Question #2 : Systems Of Trigonometric Equations

Example Question #1 : Solving Trigonometric Equations

Example Question #1 : Solving Trigonometric Equations

All Trigonometry Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: