Call Now to Set Up Tutoring:

(888) 888-0446

Precalculus : Solving Trigonometric Equations and Inequalities

Study concepts, example questions & explanations for Precalculus

Create An Account Create Tests & Flashcards

All Precalculus Resources

12 Diagnostic Tests 380 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

← Previous 1 2 Next →

Example Question #10 :Trigonometric Identities

Use trigonometric identities to solve the following equation for $\theta$ :

$tan^2\theta + sec^2\theta = 1$

Possible Answers:

$\theta = \pi/3+ n$

$\theta = \pi/4+ n\pi$

$\theta = \pi/3+ n$

$\theta= n\pi$

$\theta = \pi/2+ n\pi$

Correct answer:

$\theta= n\pi$

Explanation:

Use the trigonometric identities to switch sec into terms of tan:

$tan^2\theta + sec^2\theta = tan^2\theta + (tan^2 \theta + 1) = 2tan^2\theta + 1 = 1$

hence,

$2tan^2\theta = 0$

So we have $\tan \theta = 0$ , making $\theta= n \pi$

Therefore the solution is $\theta = n \pi$ for n being any integer.

Report an Error

Example Question #1 :Solve Trigonometric Equations And Inequalities

Which of the following is not a solution to $\small 2sin^2(x)-3sin(x)=-1$ for $\small 0\leq x<2\pi$

Possible Answers:

$\small \frac{\pi}{2}$

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

$\small \frac{\pi}{6}$

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

Correct answer:

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

Explanation:

We begin by setting the right side of the equation equal to 0.

$\small 2sin^2(x)-3sin(x)+1=0$

The equation might be easier to factor using the following substitution.

$\small w=sin(x)$

This gives the following

$\small 2w^2-3w+1=0$

This can be factored as follows

$\small \small (2w-1)(w-1)=0$

Therefore

$\small w=\frac{1}{2}\;or\;w=1$

取代我们替换因此使

$\small sin(x)=\frac{1}{2}\;or\;sin(x)=1$

Within our designated domain, we get three answers between our two equations.

$\small sin(x)=\frac{1}{2}$ when $\small x=\frac{\pi}{6}\;or\; \frac{5\pi}{6}$

$\small sin(x)=1$ when $\small x=\frac{\pi}{2}$

Therefore, the only choice that isn't correct is $\small \frac{11\pi}{6}$

Report an Error

Example Question #2 :Solve Trigonometric Equations And Inequalities

Find one possible value of.

tan(5x)+17=-55

Possible Answers:

$-446.02^{\circ}$

$0^{\circ}$

$446.02^{\circ}$

$17.84^{\circ}$

$-17.84^{\circ}$

Correct answer:

$-17.84^{\circ}$

Explanation:

Begin by isolating the tangent side of the equation:

tan(5x)+17=-55

tan(5x)=-72

Next, take the inverse tangent of both sides:

$5x=tan^{-1} (-72)$

Divide by five to get the final answer:

$x=\frac{tan^{-1} (-72)}{5}=-17.84^{\circ}$

Report an Error

Example Question #3 :Solve Trigonometric Equations And Inequalities

Use trigonometric identities to solve for the angle value.

$\tan^2\theta\sin^2\theta+\tan^2\theta\cos^2\theta+1=13$

Possible Answers:

$\theta\approx53.565$

$\theta\approx85.601$

$\theta\approx74.499$

$\theta\approx73.898$

Correct answer:

$\theta\approx73.898$

Explanation:

There are two ways to solve this problem. The first involves two trigonometric identities:

$\tan^2\theta\sin^2\theta+\tan^2\theta\cos^2\theta+1=13\\ \implies\tan^2\theta(\sin^2\theta+\cos^2\theta)+1=13\\ \implies\tan^2\theta+1=13\implies\sec^2\theta=13\\ \implies\sec\theta=\sqrt13\implies\cos\theta=\frac{\sqrt13}{13}\\ \implies \theta=\cos^{-1}\frac{\sqrt13}{13}}\approx73.898}$

The second method allows us to only use the first trigonometric identity:

$\tan^2\theta\sin^2\theta+\tan^2\theta\cos^2\theta+1=13\\ \implies\tan^2\theta(\sin^2\theta+\cos^2\theta)+1=13\\ \implies\tan^2\theta+1=13\implies\tan^2\theta=12\\ \implies\tan\theta=2\sqrt3\implies\theta=\tan^{-1}2\sqrt3\approx73.898}$

Report an Error

Example Question #1 :Solving Trigonometric Equations And Inequalities

Use trigonometric identities to solve the equation for the angle value.

$4\cos^2\theta+\cos{2\theta}+\sin^2\theta=3$

Possible Answers:

$\theta\approx39.232$

$\theta\approx78.463$

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

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

Correct answer:

$\theta\approx39.232$

Explanation:

The simplest way to solve this problem is using the double angle identity for cosine. $\cos2\theta=\cos^2\theta-\sin^2\theta$

Substituting this value into the original equation gives us:

$4\cos^2\theta+\cos^2\theta-\sin^2\theta+\sin^2\theta=3\\ \implies5\cos^2\theta=3\implies\cos^2\theta=\frac{3}{5}\\ \cos\theta=\frac{\sqrt15}{5}\implies\theta=\cos^{-1}\frac{\sqrt15}{5}}\approx39.232$

Report an Error

Example Question #5 :Solve Trigonometric Equations And Inequalities

According to the trigonometric identities, 1+cot^2(x)=?

Possible Answers:

$\frac{1}{tan(x)}$

tan(x)

$\frac{sin(x)}{cos(x)}$

csc^2(x)

Correct answer:

csc^2(x)

Explanation:

The trigonometric identity 1+cot^2(x)=csc^2(x) , is an important identity to memorize.

Some other identities that are important to know are:

$\frac{sin(x)}{cos(x)}=tan(x)$

$cot(x)=\frac{1}{tan(x)}$

sin^2(x)+cos^2(x)=1

Report an Error

Example Question #1 :Solving Trigonometric Equations And Inequalities

If $\theta$ exists in the domain from $\left[0,\frac{\pi}{2}\right]$ , solve the following: $0=1-\cos^2\theta$

Possible Answers:

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

$\pi$

$0,\pi$

$\textup{No solution.}$

Correct answer:

Explanation:

Factorize $0=1-cos^2\theta$ .

$1-cos^2\theta= (1+cos(\theta))(1-cos(\theta))$

Set both terms equal to zero and solve.

$1+cos(\theta)=0$

$cos(\theta)=-1$

$\theta=\pi$

This value is not within the $[0,\frac{\pi}{2}]$ domain.

$1-cos(\theta)=0$

$cos(\theta)=1$

$\theta=0$

This is the only correct value in the $[0,\frac{\pi}{2}]$ domain.

Report an Error

Example Question #1 :Solving Trigonometric Equations And Inequalities

Solve for $\theta$ in the equation $sin^2(\theta)-2sin(\theta)+1=0$ on the interval $0\leq\theta\leq2\pi$ .

Possible Answers:

$\theta=\frac{3\pi}{4}$

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

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

$\theta=0$

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

Correct answer:

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

Explanation:

If you substitute $x=sin(\theta)$ you obtain a recognizable quadratic equation which can be solved for,

(x-1)^2=0 .

Then we can plugback into our equation and use the unit circle to find that

$\theta=\frac{\pi}{2}$ .

Report an Error

Example Question #1 :Solving Trigonometric Equations And Inequalities

Given that theta exists from $[0,2\pi]$ , solve: $2cos^2\theta=cos(\theta)$

Possible Answers:

$\textup{None of the given answers.}$

$\frac{\pi}{2}$

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

$\frac{\pi}{3},\frac{\pi}{2},\frac{3\pi}{2}, \frac{5\pi}{3}$

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

Correct answer:

$\frac{\pi}{3},\frac{\pi}{2},\frac{3\pi}{2}, \frac{5\pi}{3}$

Explanation:

In order to solve $2cos^2\theta=cos(\theta)$ appropriately, do not divide $cos(\theta)$ on both sides. The effect will eliminate one of the roots of this trig function.

Substract $cos(\theta)$ from both sides.

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

Factor the left side of the equation.

$cos(\theta)[2cos(\theta)-1]=0$

每一项等于零,解出θwith the restriction $[0,2\pi]$ .

$cos(\theta)=0$

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

$2cos(\theta)-1=0$

$2cos(\theta)=1$

$cos(\theta)=\frac{1}{2}$

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

The correct answer is:

$\theta=\frac{\pi}{2},\frac{3\pi}{2},\frac{\pi}{3}, \frac{5\pi}{3}$

Report an Error

Example Question #2 :Solving Trigonometric Equations And Inequalities

Solve $\sin^2(x) + 4 = 0$ for $0^{\circ}\leq x\leq 360^{\circ}$

Possible Answers:

x = -2, x = 2

There is no solution.

x = 0

$x = 90^{\circ}$

Correct answer:

There is no solution.

Explanation:

By subtractingfrom both sides of the original equation, we get $\sin^2(x) = -4$ . We know that the square of a trigonometric identity cannot be negative, regardless of the input, so there can be no solution.

Report an Error

← Previous 1 2 Next →

View Pre-Calculus Tutors

Ranjeeta
Certified Tutor

University of Rajasthan, Bachelor of Science, Mathematics. University of Rajasthan, Master of Science, Mathematics.

View Pre-Calculus Tutors

Elisa
Certified Tutor

Cedarville University, Bachelor in Arts, Mathematics Teacher Education.

View Pre-Calculus Tutors

Liban
Certified Tutor

Emory University, Bachelor of Science, Mathematics/Economics.

All Precalculus Resources

12 Diagnostic Tests 380 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Computer Science Tutors in Phoenix,MCAT Tutors in Houston,ACT Tutors in Denver,MCAT Tutors in San Diego,MCAT Tutors in Dallas Fort Worth,Reading Tutors in Los Angeles,Algebra Tutors in Chicago,数学老师在凤凰城,LSAT Tutors in San Diego,LSAT Tutors in Houston

Popular Courses & Classes

SSAT Courses & Classes in Chicago,ACT Courses & Classes in Los Angeles,SAT Courses & Classes in New York City,GRE Courses & Classes in Philadelphia,SAT Courses & Classes in Washington DC,GMAT Courses & Classes in Chicago,SAT Courses & Classes in San Diego,Spanish Courses & Classes in Dallas Fort Worth,SAT Courses & Classes in Philadelphia,SSAT Courses & Classes in Boston

Popular Test Prep

LSAT Test Prep in Philadelphia,GMAT Test Prep in Boston,ISEE Test Prep in Denver,MCAT Test Prep in New York City,MCAT Test Prep in Los Angeles,SAT Test Prep in Denver,GMAT Test Prep in Chicago,SAT Test Prep in Houston,ACT Test Prep in Seattle,SAT Test Prep in Seattle

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.

我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

Precalculus : Solving Trigonometric Equations and Inequalities

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #10 :Trigonometric Identities

Example Question #1 :Solve Trigonometric Equations And Inequalities

Example Question #2 :Solve Trigonometric Equations And Inequalities

Example Question #3 :Solve Trigonometric Equations And Inequalities

Example Question #1 :Solving Trigonometric Equations And Inequalities

Example Question #5 :Solve Trigonometric Equations And Inequalities

Example Question #1 :Solving Trigonometric Equations And Inequalities

Example Question #1 :Solving Trigonometric Equations And Inequalities

Example Question #1 :Solving Trigonometric Equations And Inequalities

Example Question #2 :Solving Trigonometric Equations And Inequalities

All Precalculus Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Find the Best Tutors

Email address:
Your name:
Feedback: