Call Now to Set Up Tutoring:

(888) 888-0446

Common Core: High School - Geometry : Prove Laws of Sines and Cosines: CCSS.Math.Content.HSG-SRT.D.10

Study concepts, example questions & explanations for Common Core: High School - Geometry

Create An Account

All Common Core: High School - Geometry Resources

6 Diagnostic Tests 114 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Prove Laws Of Sines And Cosines: Ccss.Math.Content.Hsg Srt.D.10

The Law of Cosines is a generalization of which common theorem?

Possible Answers:

Pythagorean Theorem

Triangle Sum Theorem

Side Angle Side Similarity Theorem

Triangle Proportionality Theorem

Fundamental Theorem of Algebra

Correct answer:

Pythagorean Theorem

Explanation:

The Law of Cosines is a generalization of the Pythagorean Theorem. To better understand this, start by drawing triangle ABC with sides a, b, and c. Then, construct the altitude from point A to get height h. Finally label the distance between point B and the point where h meets side a as distance r.

Screen shot 2020 07 01 at 7.58.05 pm

We will arbitrarily work to solve for side b, but we could do the same for a and c (after drawing the correct altitudes), which is why we can generalize that the other two formulas to find a and c follow.

By definition, we know that $\sin\angle B= \frac{h}{c}$ . If we multiply both sides by c, we get $c\sin\angle B= h$ .

Also, we know that $\cos\angle B= \frac{r}{c}$ . If we multiply both sides by c, we get $c\cos\angle B= r$ .

Using the Pythagorean Theorem, we get b^2=h^2+(a-r)^2 .

Substituting in h and r from above, we get:

$b^2=(c\sin \angle B)^2+ (a-c\cos\angle B)^2$

$= c^2\sin^2 \angle B + a^2-2ac\cos\angle B + c^2\cos^2\angle B$

$= c^2 (sin^2 \angle B+cos^2\angle B) + a^2-2ac\cos\angle B$ (because $sin^2 \angle B+cos^2\angle B=1$ )

$= c^2 + a^2-2ac\cos\angle B$

Therefore $b^2= c^2 + a^2-2ac\cos\angle B$ .

The other two versions of Law of Cosines are:

$a^2= b^2 + c^2-2bc\cos\angle A$

$c^2= a^2 + b^2-2ab\cos\angle C$

Report an Error

Example Question #2 : Prove Laws Of Sines And Cosines: Ccss.Math.Content.Hsg Srt.D.10

True or false: The Law of Sines applies to all types of triangles.

Possible Answers:

True

False

Correct answer:

True

Explanation:

This is true. While the Pythagorean Theorem only applies to right triangles, the Law of Sines applies to all triangles no matter if they are right, acute, or obtuse.

Report an Error

Example Question #1 : Prove Laws Of Sines And Cosines: Ccss.Math.Content.Hsg Srt.D.10

True or False: The Law of Cosines applies to all triangles.

Possible Answers:

True

False

Correct answer:

True

Explanation:

This is true. While the Pythagorean Theorem only applies to right triangles, the Law of Cosines applies to all triangles no matter if they are right, acute, or obtuse.

Report an Error

Example Question #4 : Prove Laws Of Sines And Cosines: Ccss.Math.Content.Hsg Srt.D.10

Choose the answer choice that correctly states the Law of Sines, given the triangle below.

Screen shot 2020 08 19 at 4.54.09 pm

Possible Answers:

$\frac{A}{\sin (\gamma )}=\frac{B}{\sin (\alpha )}=\frac{C}{\sin (\beta )}$

$A^{2}+B^{2}-2AB\sin (\gamma )=C^{2}$

$A^{2}+B^{2}-2AB\cos (\gamma )=C^{2}$

$\frac{A}{\cos (\gamma )}=\frac{B}{\cos (\alpha )}=\frac{C}{\cos (\beta )}$

Correct answer:

$\frac{A}{\sin (\gamma )}=\frac{B}{\sin (\alpha )}=\frac{C}{\sin (\beta )}$

Explanation:

The Law of Sines uses ratios of a triangle’s sides and angles to allow us to be able to solve for unknown sides and/or angles. This relationship is true for any triangle.

Screen shot 2020 08 19 at 4.54.09 pm

The Law of Sines is $\frac{A}{\sin (\gamma )}=\frac{B}{\sin (\alpha )}=\frac{C}{\sin (\beta )}$ .

Report an Error

Example Question #5 : Prove Laws Of Sines And Cosines: Ccss.Math.Content.Hsg Srt.D.10

Which of the following expressions correctly states the Law of Cosines? Use the following triangle for reference.

Screen shot 2020 08 20 at 10.56.17 am

Possible Answers:

$\frac{A}{\sin (\gamma )}=\frac{B}{\sin (\alpha )}=\frac{C}{\sin (\beta )}$

$A^{2}+B^{2}-2AB\sin (\gamma )=C^{2}$

$A^{2}+B^{2}-2AB\cos (\gamma )=C^{2}$

$\frac{A}{\cos (\gamma )}=\frac{B}{\cos (\alpha )}=\frac{C}{\cos (\beta )}$

Correct answer:

$A^{2}+B^{2}-2AB\cos (\gamma )=C^{2}$

Explanation:

The Law of Cosines should look similar to a theorem we use quite commonly for right triangles. The Law of Cosines is basically the Pythagorean Theorem, but it has an added argument so that it is true for any and all triangles.

Screen shot 2020 08 20 at 10.56.17 am

The Law of Cosines is $A^{2}+B^{2}-2AB\cos (\gamma )=C^{2}$

Report an Error

Example Question #6 : Prove Laws Of Sines And Cosines: Ccss.Math.Content.Hsg Srt.D.10

Choose the answer choice that correctly states the Law of Sines, given the triangle below.

Screen shot 2020 08 20 at 11.04.58 am

Possible Answers:

$A^{2}+B^{2}-2AB\cos (\gamma )=C^{2}$

$A^{2}+B^{2}-2AB\sin (\gamma )=C^{2}$

$\frac{A}{\cos (\gamma )}=\frac{B}{\cos (\alpha )}=\frac{C}{\cos (\beta )}$

$\frac{A}{\sin (\gamma )}=\frac{B}{\sin (\alpha )}=\frac{C}{\sin (\beta )}$

Correct answer:

$\frac{A}{\sin (\gamma )}=\frac{B}{\sin (\alpha )}=\frac{C}{\sin (\beta )}$

Explanation:

The Law of Sines uses ratios of a triangle’s sides and angles to allow us to be able to solve for unknown sides and/or angles. This relationship is true for any triangle.

Screen shot 2020 08 20 at 11.04.58 am

The Law of Sines is $\frac{A}{\sin (\gamma )}=\frac{B}{\sin (\alpha )}=\frac{C}{\sin (\beta )}$ .

Report an Error

View Tutors

Anne
Certified Tutor

University of Maryland-Baltimore, Doctor of Science, Accounting.

View Tutors

Satweka
Certified Tutor

The University of Texas at Dallas, Master of Science, Biomedical Engineering.

View Tutors

Tonja
Certified Tutor

McNeese State University, Bachelor in Arts, English.

All Common Core: High School - Geometry Resources

6 Diagnostic Tests 114 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Physics Tutors in Dallas Fort Worth, MCAT Tutors in San Diego, Statistics Tutors in Miami, Reading Tutors in New York City, French Tutors in Los Angeles, LSAT Tutors in Houston, Algebra Tutors in Phoenix, GRE Tutors in San Diego, GMAT Tutors in Boston, Calculus Tutors in Los Angeles

Popular Courses & Classes

SAT Courses & Classes in Houston, LSAT Courses & Classes in Dallas Fort Worth, ISEE Courses & Classes in Washington DC, MCAT Courses & Classes in Denver, GRE Courses & Classes in Seattle, ISEE Courses & Classes in Boston, LSAT Courses & Classes in Houston, Spanish Courses & Classes in San Diego, GMAT Courses & Classes in Los Angeles, Spanish Courses & Classes in New York City

Popular Test Prep

GMAT Test Prep in Miami, LSAT Test Prep in Boston, GMAT Test Prep in Los Angeles, SSAT Test Prep in Los Angeles, ACT Test Prep in Denver, ISEE Test Prep in Washington DC, SSAT Test Prep in Atlanta, ISEE Test Prep in Houston, GMAT Test Prep in San Diego, GRE Test Prep in Chicago

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

Common Core: High School - Geometry : Prove Laws of Sines and Cosines: CCSS.Math.Content.HSG-SRT.D.10

Study concepts, example questions & explanations for Common Core: High School - Geometry

All Common Core: High School - Geometry Resources

Example Questions

Example Question #1 : Prove Laws Of Sines And Cosines: Ccss.Math.Content.Hsg Srt.D.10

Example Question #2 : Prove Laws Of Sines And Cosines: Ccss.Math.Content.Hsg Srt.D.10

Example Question #1 : Prove Laws Of Sines And Cosines: Ccss.Math.Content.Hsg Srt.D.10

Example Question #4 : Prove Laws Of Sines And Cosines: Ccss.Math.Content.Hsg Srt.D.10

Example Question #5 : Prove Laws Of Sines And Cosines: Ccss.Math.Content.Hsg Srt.D.10

Example Question #6 : Prove Laws Of Sines And Cosines: Ccss.Math.Content.Hsg Srt.D.10

All Common Core: High School - Geometry Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: