We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

Trigonometry : Factoring Trigonometric Equations

Study concepts, example questions & explanations for Trigonometry

Create An Account Create Tests & Flashcards

All Trigonometry Resources

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

Example Questions

← Previous 1 2 Next →

Example Question #1 :Factoring Trigonometric Equations

Factor $1-2\cos x+\cos^2 x$ .

Possible Answers:

$\sin^2x$

$\sin^4 x$

$(1-\cos x)^2$

$\cos^4 x$

$\sin x \tan x$

Correct answer:

$(1-\cos x)^2$

Explanation:

Don't get scared off by the fact we're doing trig functions! Factor as you normally would. Because our middle term is negative ( $-2\cos x$ ), we know that the signs inside of our parentheses will be negative.

This means that $1-2\cos x+\cos^2 x$ can be factored to $(1-\cos x)(1-\cos x)$ or $(1-\cos x)^2$ .

Report an Error

Example Question #2 :Factoring Trigonometric Equations

Which of the following values ofin radians satisfy the equation

tan^2(x)-tan(x)=0

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

Possible Answers:

1, 2, and 3

1 only

1 and 2

3 only

2 only

Correct answer:

1 only

Explanation:

The fastest way to solve this equation is to simply try the three answers. Plugging in $\frac{\pi}{4}$ gives

$tan^2(\frac{\pi}{4})-tan(\frac{\pi}{4})=(1)^2-1=1-1=0$

Our first choice is valid.

Plugging in $\frac{\pi}{2}$ gives

$tan^2(\frac{\pi}{2})-tan(\frac{\pi}{2})$

However, since $tan(\frac{\pi}{}2)$ is undefined, this cannot be a valid answer.

Finally, plugging in $\frac{3\pi}{4}$ gives

$tan^2(\frac{3\pi}{4})-tan(\frac{3\pi}{4})=(-1)^2-(-1)=1+1=2$

Therefore, our third answer choice is not correct, meaning only 1 is correct.

Report an Error

Example Question #3 :Factoring Trigonometric Equations

$f(x)=\sin^2(x)+\cos^2(x)-\sin(2x)$

Find the zeros of the above equation in the interval

$[0,\pi]$ .

Possible Answers:

$\frac{\pi}{4}$

$\pi$

$\frac{\pi}{2}$

$2\pi$

Correct answer:

$\frac{\pi}{4}$

Explanation:

$f(x)=\sin^2(x)-\sin(2x)+\cos^2(x)=1-2\sin(x)cos(x)$

$=1-2\sin(x)cos(x)=0$

Therefore,

1=2sin(x)cos(x)

$\frac{1}{2}=sin(x)cos(x)$

and that only happens once in the given interval, at $\frac{\pi}{4}$ , or 45 degrees.

$\sin\bigg(\frac{\pi}{4}\bigg)\cos\bigg(\frac{\pi}{4}\bigg)=\frac{1}{\sqrt2}\cdot \frac{1}{\sqrt2}=\frac{1}{2}$

Report an Error

Example Question #4 :Factoring Trigonometric Equations

Factor the expression

1+2cos(x)sin(x)

Possible Answers:

$(cos(x)+sin(x))^{2}$

$(cos(x)-sin(x))^{2}$

$(cos(x)+2sin(x))^{2}$

$2(cos(x)+sin(x))^{2}$

$(cos(x)+sin(x))^{3}$

Correct answer:

$(cos(x)+sin(x))^{2}$

Explanation:

We have cos^2x+sin^2x=1 .

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

This last expression can be written as :

(cos(x)+sin(x))^2 .

This shows the required result.

Report an Error

Example Question #1 :Factoring Trigonometric Equations

Factor the following expression:

$sin^{3}x-1$

Possible Answers:

(sinx+1)(sin^2x+sinx+1)

(sinx-1)(sin^2x+2sinx+1)

(sinx-1)(sin^2x+sinx+1)

(2sinx-1)(sin^2x+sinx+1)

(sinx-1)(sin^2x+sinx-1)

Correct answer:

(sinx-1)(sin^2x+sinx+1)

Explanation:

We know that we can write

a^3-1 in the following form

(a^3-1)=(a-1)(a^2+a+1) .

Now taking a=sinx ,

we have:

sin^3 x-1=(sinx-1)(sin^2 x+sinx+1) .

This is the result that we need.

Report an Error

Example Question #6 :Factoring Trigonometric Equations

We accept that :

cos(2x)=cos^2x-sin^2x

What is a simple expression of

$cos^{4}x-sin^{4}x$

Possible Answers:

cos2x

-sin2x

2cos2x

-2cos2x

2sin(2x)

Correct answer:

cos2x

Explanation:

First we see that :

a^4-b^4=(a^2+b^2)(a^2-b^2) .

Now letting a=cosx ,b=sinx

we have

cos^4x-sin^4x=(cos^2x+sin^2x)(cos^2x-sin^2x)

We know that :

cos^2x+sin^2x=1 and we are given that

cos^2x-sin^2x=cos2x , this gives

cos^4x-sin^4x=cos2x

Report an Error

Example Question #7 :Factoring Trigonometric Equations

Factor the following expression:

cos^8x-sin^8x

Possible Answers:

(2cos^4x+sin^4x)(cos^2x-sin^2x)

(cos^4x+2sin^4x)(cos^2x-sin^2x)

(cos^4x+sin^4x)(cos^2x-sin^2x)

We can't factor this expression.

(cos^4x+sin^4x)(-cos^2x-sin^2x)

Correct answer:

(cos^4x+sin^4x)(cos^2x-sin^2x)

Explanation:

Note first that:

a^8-b^8=(a^4-b^4)(a^4+b^4) and :

(a^4-b^4)=(a^2+b^2)(a^2 - b^2) .

Now taking $a=cosx \quad b=sinx$ . We have

cos^8x-sin^8x=(cos^4x+sin^4x)(cos^4-sin^4x) .

Since cos^4x-sin^4x=(cos^2x+sin^2x)(cos^2x-sin^2x) and cos^2x+sin^2x=1 .

We therefore have :

cos^8x-sin^8x=(cos^4x+sin^4x)(cos^2x-sin^2x)

Report an Error

Example Question #8 :Factoring Trigonometric Equations

Factor the following expression

$sin^{2n}(x)+2sin^n(x)+3$ whereis assumed to be a positive integer.

Possible Answers:

(sin^nx-1)(sin^nx-2)

(sin^nx+1)(sin^nx+2)

(sin^nx+1)(sin^nx-2)

(2sin^nx+1)(sin^nx-2)

We cannot factor the above expression.

Correct answer:

We cannot factor the above expression.

Explanation:

Letting sin^n(x)=X , we have the equivalent expression:

X^2+2X+3 .

We cant factor X^2+2X+3 since $X^2+2X+3=(x+1)^2+2 >0 \forall \quadx\in\mathbb{R}$ .

This shows that we cannot factor the above expression.

Report an Error

Example Question #1 :Factoring Trigonometric Equations

Factor

$1-cos^{11}x$

Possible Answers:

$(1-cosx)(cos^{10}x+cos^{9}x+...cosx-1)$

$(1-cosx)(cos^{10}x+cos^{9}x+...cosx+1)$

$(1-cosx)(cos^{11}x+cos^{9}x+...cosx+1)$

$(2-cosx)(cos^{10}x+cos^{9}x+...cosx+1)$

$(-1-cosx)(cos^{10}x+cos^{9}x+...cosx+1)$

Correct answer:

$(1-cosx)(cos^{10}x+cos^{9}x+...cosx+1)$

Explanation:

We first note that we have:

$1-a^{11}=(1-a)(a^{10}+a^9+...a+1)$

Then taking a=cos(x) , we have the result.

$(1-cosx)(cos^{10}x+cos^{9}x+...cosx+1)$

Report an Error

Example Question #10 :Factoring Trigonometric Equations

Find a simple expression for the following :

$cos^{2n}x-cos^{2n+2} x$

Possible Answers:

3sinx

$-cos^{2n}xsin^{2}x$

$cos^{2n}x (1-sin^{2}x)$

$cos^{2n}xsin^{2}x$

$2cos^{2n}xsin^{2}x$

Correct answer:

$cos^{2n}xsin^{2}x$

Explanation:

First of all we know that :

cos^2x+sin^2x=1 and this gives:

$1-cos^2{x}=sin^{2}x$ .

Now we need to see that: $cos^{2n}x-cos^{2n+2} x$ can be written as

$cos^{2n}x(1-cos^2x)$ and since $1-cos^2{x}=sin^{2}x$

we have then:

$cos^{2n}x(1-cos^2x)=cos^{2n}xsin^2x$ .

Report an Error

← Previous 1 2 Next →

View Trigonometry Tutors

Brianna
Certified Tutor

Fairfield University, Bachelor of Science, Biology, General. National University of Ireland at Galway, Master of Science, Neu...

View Trigonometry Tutors

John
Certified Tutor

Iowa State University, Bachelor of Science, Applied Physics.

View Trigonometry Tutors

Kaitlyn
Certified Tutor

Fairfield University, Bachelor of Science, Biology, General. NUI Galway Ireland, Master of Science, Neuroscience.

All Trigonometry Resources

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

Trigonometry Tutoring in Top Cities:

Trigonometry Tutors in Top Cities:

Popular Courses & Classes

GRE Courses & Classes in Dallas Fort Worth,ISEE Courses & Classes in Houston,GMAT Courses & Classes in Chicago,SSAT Courses & Classes in Boston,ISEE Courses & Classes in Los Angeles,GMAT Courses & Classes in New York City,ISEE Courses & Classes in Seattle,SSAT Courses & Classes in Houston,Spanish Courses & Classes in Houston,LSAT Courses & Classes in Los Angeles

流行的测试准备

ACT Test Prep in Los Angeles,SSAT Test Prep in San Diego,ACT Test Prep in San Diego,GRE Test Prep in Seattle,GRE Test Prep in San Diego,SAT Test Prep in Denver,GMAT Test Prep in Denver,GMAT Test Prep in New York City,ACT Test Prep in Atlanta,GRE 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.

请建议你将dama负责ges (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

我是老板,或代理授权采取行动ehalf 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 : Factoring Trigonometric Equations

Study concepts, example questions & explanations for Trigonometry

All Trigonometry Resources

Example Questions

Example Question #1 :Factoring Trigonometric Equations

Example Question #2 :Factoring Trigonometric Equations

Example Question #3 :Factoring Trigonometric Equations

Example Question #4 :Factoring Trigonometric Equations

Example Question #1 :Factoring Trigonometric Equations

Example Question #6 :Factoring Trigonometric Equations

Example Question #7 :Factoring Trigonometric Equations

Example Question #8 :Factoring Trigonometric Equations

Example Question #1 :Factoring Trigonometric Equations

Example Question #10 :Factoring Trigonometric Equations

All Trigonometry Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Find the Best Tutors

Email address:
Your name:

Email address:
Your name:
Feedback: