We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

ACT Math : How to find a reference angle

Study concepts, example questions & explanations for ACT Math

Create An Account

All ACT Math Resources

14 Diagnostic Tests 767 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : How To Find A Reference Angle

Which of the following is equivalent to cot(θ)sec(θ)sin(θ)

Possible Answers:

cot(θ)

–sec(θ)

tan(θ)

Correct answer:

Explanation:

The first thing to do is to breakdown the meaning of each trig function, cot = cos/sin, sec = 1/cos, and sin = sin. Then put these back into the function and simplify if possible, so then (cos (Θ)/sin (Θ))*(1/cos (Θ))*(sin (Θ)) = (cos (Θ)*sin(Θ))/(sin (Θ)*cos(Θ)) = 1, since they all cancel out.

Report an Error

Example Question #2 : Reference Angles

Using trigonometry identities, simplify sinθcos²θ – sinθ

Possible Answers:

cos²θsinθ

–sin³θ

None of these answers are correct

sin²θcosθ

cos³θ

Correct answer:

–sin³θ

Explanation:

Factor the expression to get sinθ(cos²θ – 1).

The trig identity cos²θ + sin²θ = 1 can be reworked to becomes cos²θ – 1 = –sinθ resulting in the simplification of –sin³θ.

Report an Error

Example Question #2 : How To Find A Reference Angle

Using trig identities, simplify sinθ + cotθcosθ

Possible Answers:

sin²θ

cos²θ

tanθ

cscθ

secθ

Correct answer:

cscθ

Explanation:

Cotθ can be written as cosθ/sinθ, which results in sinθ + cos²θ/sinθ.

Combining to get a single fraction results in (sin²θ + cos²θ)/sinθ.

Knowing that sin²θ + cos²θ = 1, we get 1/sinθ, which can be written as cscθ.

Report an Error

Example Question #4 : Reference Angles

Simplify sec⁴Θ – tan⁴Θ.

Possible Answers:

sinΘ + cosΘ

secΘ + sinΘ

sec²Θ + tan²Θ

tan²Θ – sin²Θ

cosΘ – sinΘ

Correct answer:

sec²Θ + tan²Θ

Explanation:

Factor using the difference of two squares: a² – b² = (a + b)(a – b)

The identity 1 + tan²Θ = sec²Θ which can be rewritten as 1 = sec²Θ – tan²Θ

So sec⁴Θ – tan⁴Θ = (sec²Θ + tan²Θ)(sec²Θ – tan²Θ) = (sec²Θ + tan²Θ)(1) = sec²Θ + tan²Θ

Report an Error

Example Question #1 : How To Find A Reference Angle

Evaluate the expression below.

$\displaystyle sin(45^o) + cot(45^o)$

Possible Answers:

$\frac{2 + \sqrt{2}}{2}$ $\displaystyle \frac{2 + \sqrt{2}}{2}$

$\frac{1 + \sqrt{2}}{2}$ $\displaystyle \frac{1 + \sqrt{2}}{2}$

$\sqrt{2}$ $\displaystyle \sqrt{2}$

$\frac{2 + \sqrt{3}}{2}$ $\displaystyle \frac{2 + \sqrt{3}}{2}$

$\frac{1 + \sqrt{3}}{2}$ $\displaystyle \frac{1 + \sqrt{3}}{2}$

Correct answer:

$\frac{2 + \sqrt{2}}{2}$ $\displaystyle \frac{2 + \sqrt{2}}{2}$

Explanation:

At 45^o $\displaystyle 45^o$, sine and cosine have the same value.

$sin(45^o)=cos(45^o)=\frac{\sqrt{2}}{2}$ $\displaystyle sin(45^o)=cos(45^o)=\frac{\sqrt{2}}{2}$

Cotangent is given by $\frac{cos}{sin}$ $\displaystyle \frac{cos}{sin}$.

$cot(45^o=\frac{cos(45^o)}{sin(45^o)})=\frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}}=1$ $\displaystyle cot(45^o=\frac{cos(45^o)}{sin(45^o)})=\frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}}=1$

Now we can evaluate the expression.

$sin(45^o)+cot(45^o)=(\frac{\sqrt{2}}{2})+1=(\frac{\sqrt{2}}{2})+\frac{2}{2}=\frac{\sqrt{2}+2}{2}$ $\displaystyle sin(45^o)+cot(45^o)=(\frac{\sqrt{2}}{2})+1=(\frac{\sqrt{2}}{2})+\frac{2}{2}=\frac{\sqrt{2}+2}{2}$

Report an Error

Example Question #6 : Reference Angles

What is the reference angle of an angle that measures 3510 in standard position?

Possible Answers:

109

351

369

Correct answer:

Explanation:

3600 – 3510 = 90

Report an Error

Example Question #7 : Reference Angles

Simplify the following expression:

$\frac{sin\theta}{cot\theta sec\theta}$ $\displaystyle \frac{sin\theta}{cot\theta sec\theta}$

Possible Answers:

None of the answers are correct

tanΘ

cscΘ

sin²Θ

cos²Θ

Correct answer:

sin²Θ

Explanation:

Convert cotΘ and secΘ to sinΘ and cosΘ and simplify the resulting complex fraction.

cotΘ = cosΘ secΘ = 1

sinΘ cosΘ

Report an Error

Example Question #2 : How To Find A Reference Angle

What is the reference angle for $855^{\circ}$ $\displaystyle 855^{\circ}$?

Possible Answers:

$495^{\circ}$ $\displaystyle 495^{\circ}$

$55^{\circ}$ $\displaystyle 55^{\circ}$

$360^{\circ}$ $\displaystyle 360^{\circ}$

$720^{\circ}$ $\displaystyle 720^{\circ}$

$45^{\circ}$ $\displaystyle 45^{\circ}$

Correct answer:

$45^{\circ}$ $\displaystyle 45^{\circ}$

Explanation:

The reference angle is between 0^o $\displaystyle 0^o$ and $90^{\circ}$ $\displaystyle 90^{\circ}$, starting on the positive $\displaystyle x$-axis and rotating in a counter-clockwise manor.

To find the reference angle, we subtract 360^o $\displaystyle 360^o$ for each complete revolution until we get a positive number less than 360^o $\displaystyle 360^o$.

$\displaystyle 855 - 360 = 495$

$\displaystyle 495 - 360 = 135$

855^o $\displaystyle 855^o$ is equal to two complete revolutions, plus a $135^{\circ}$ $\displaystyle 135^{\circ}$ angle. Since $135^{\circ}$ $\displaystyle 135^{\circ}$ is in Quadrant II, we subtract it from $180^{\circ}$ $\displaystyle 180^{\circ}$ to get our reference angle:

$180-135=45^{\circ}$ $\displaystyle 180-135=45^{\circ}$

Report an Error

Example Question #5 : How To Find A Reference Angle

Unit_circle

In the unit circle above, if $\Theta =30^{\circ}$ $\displaystyle \Theta =30^{\circ}$, what are the coordinates of (X,Y) $\displaystyle (X,Y)$?

Possible Answers:

$\left (\frac{\sqrt{3}}{2},\frac{1}{2} \right )$ $\displaystyle \left (\frac{\sqrt{3}}{2},\frac{1}{2} \right )$

$\left (\frac{\sqrt{3}}{2},\frac{\sqrt{3}}{2} \right )$ $\displaystyle \left (\frac{\sqrt{3}}{2},\frac{\sqrt{3}}{2} \right )$

$\left (\frac{1}{2},\frac{\sqrt{3}}{2} \right )$ $\displaystyle \left (\frac{1}{2},\frac{\sqrt{3}}{2} \right )$

$\left (1,\frac{1}{2} \right )$ $\displaystyle \left (1,\frac{1}{2} \right )$

$\left (\frac{1}{2},\frac{1}{2} \right )$ $\displaystyle \left (\frac{1}{2},\frac{1}{2} \right )$

Correct answer:

$\left (\frac{\sqrt{3}}{2},\frac{1}{2} \right )$ $\displaystyle \left (\frac{\sqrt{3}}{2},\frac{1}{2} \right )$

Explanation:

On the unit circle, (X,Y) = (cos Θ, sin Θ).

(cos Θ,sin Θ) = (cos 30º, sin 30º) = (√3 / 2 , 1 / 2)

Report an Error

Example Question #4 : Reference Angles

What is the reference angle for $45^{\circ}$ $\displaystyle 45^{\circ}$?

Possible Answers:

$180^{\circ}$ $\displaystyle 180^{\circ}$

$90^{\circ}$ $\displaystyle 90^{\circ}$

$135^{\circ}$ $\displaystyle 135^{\circ}$

$45^{\circ}$ $\displaystyle 45^{\circ}$

$315^{\circ}$ $\displaystyle 315^{\circ}$

Correct answer:

$45^{\circ}$ $\displaystyle 45^{\circ}$

Explanation:

A reference angle is the smallest possible angle between a given angle measurement and the x-axis.

In this case, our angle $45^{\circ}$ $\displaystyle 45^{\circ}$ lies in Quadrant I, so the angle is its own reference angle.

Thus, the reference angle for $45^{\circ}$ $\displaystyle 45^{\circ}$ is $45^{\circ}$ $\displaystyle 45^{\circ}$.

Report an Error

View ACT Math Tutors

Aiay
Certified Tutor

The University of Texas at Austin, Bachelor of Engineering, Electrical Engineering.

View ACT Math Tutors

Noah
Certified Tutor

Bard College, Bachelors, Philosophy and Creative Writing (Dual Major). Sarah Lawrence College, Master of Fine Arts, Creative ...

View ACT Math Tutors

Ping
Certified Tutor

The University of Texas at Dallas, Bachelor of Science, Computer Science. University of North Texas, Master of Arts, Education.

All ACT Math Resources

14 Diagnostic Tests 767 Practice Tests Question of the Day Flashcards Learn by Concept

ACT Math Tutors in Top Cities:

Atlanta ACT Math Tutors, Austin ACT Math Tutors, Boston ACT Math Tutors, Chicago ACT Math Tutors, Dallas Fort Worth ACT Math Tutors, Denver ACT Math Tutors, Houston ACT Math Tutors, Kansas City ACT Math Tutors, Los Angeles ACT Math Tutors, Miami ACT Math Tutors, New York City ACT Math Tutors, Philadelphia ACT Math Tutors, Phoenix ACT Math Tutors, San Diego ACT Math Tutors, San Francisco-Bay Area ACT Math Tutors, Seattle ACT Math Tutors, St. Louis ACT Math Tutors, Tucson ACT Math Tutors, Washington DC ACT Math Tutors

Popular Courses & Classes

SSAT Courses & Classes in New York City, SSAT Courses & Classes in Los Angeles, LSAT Courses & Classes in Miami, MCAT Courses & Classes in Dallas Fort Worth, GRE Courses & Classes in New York City, Spanish Courses & Classes in Houston, LSAT Courses & Classes in Seattle, MCAT Courses & Classes in Washington DC, LSAT Courses & Classes in Denver, SSAT Courses & Classes in Philadelphia

Popular Test Prep

GRE Test Prep in Denver, MCAT Test Prep in New York City, GRE Test Prep in Dallas Fort Worth, SAT Test Prep in Chicago, LSAT Test Prep in Seattle, GMAT Test Prep in Boston, LSAT Test Prep in Chicago, SAT Test Prep in Phoenix, SSAT Test Prep in Miami, ACT Test Prep in Dallas Fort Worth

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

ACT Math : How to find a reference angle

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #1 : How To Find A Reference Angle

Example Question #2 : Reference Angles

Example Question #2 : How To Find A Reference Angle

Example Question #4 : Reference Angles

Example Question #1 : How To Find A Reference Angle

Example Question #6 : Reference Angles

Example Question #7 : Reference Angles

Example Question #2 : How To Find A Reference Angle

Example Question #5 : How To Find A Reference Angle

Example Question #4 : Reference Angles

All ACT Math Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: