We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

Trigonometry : Law of Cosines

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

← Previous 1 2 3 4 Next →

Example Question #311 : Trigonometry

In degrees, find the value of angle A.

Possible Answers:

115.61

109.04

108.21

102.56

Correct answer:

108.21

Explanation:

In order to find the value of the angle, you will need to use the law of cosines. Recall that for any triangle, like the one shown below,

$\cos A=\frac{b^2+c^2-a^2}{2bc}$

$\cos B=\frac{a^2+c^2-b^2}{2ac}$

$\cos C=\frac{a^2+b^2-c^2}{2ab}$

Now, since we want to find the value of A, we will need to use $\cos A=\frac{b^2+c^2-a^2}{2bc}$ .

Plug in the given values of the triangle.

$\cos A=\frac{2^2+4^2-5^2}{2(2)(4)}=-0.3125$

A=108.21 degrees

Make sure to round to places after the decimal.

Report an Error

Example Question #51 : Triangles

In triangle $\triangle ABC$ , $\measuredangle A = 115^{\circ}$ , AB = 31 and AC = 42 . To the nearest tenth, what is {}BC ?

Possible Answers:

61.9

71.3

57.3

3825.5

40.3

Correct answer:

61.9

Explanation:

By the Law of Cosines,

$BC^2 = AB^2 +AC^2 - 2\left (AB\right )\left ( AC\right )\cos\angle A$

or, equivalently,

$BC = \sqrt[]{AB^2 +AC^2 - 2\left ( AB\right )\left ( AC\right )\cos \angle A}$

Substitute:

$BC = \sqrt[]{31^2 + 42^2 - 2\left ( 31\right )\left ( 42\right )\cos \left ( 115^{\circ}\right )}$

$\textup{BC}= \sqrt[]{961 + 1764 - 2604\cdot (-0.4226)} \approx \sqrt{2725 + 1100} \approx \sqrt{3825}\approx61.9$

Report an Error

Example Question #314 : Trigonometry

Solve for $\theta$ :

Cosines 2

Possible Answers:

83.04 ^o

41.41 ^o

124.23^o

14.36^o

88.57^o

Correct answer:

41.41 ^o

Explanation:

Solve using law of cosines: $c^2 = a^2 + b^2 - 2ab \cdot \cos C$ where C is the angle between sides a and b.

$8^2 = 3^2 + 10^2 - 2(3)(10) \cos \theta$

$64 = 109 - 60 \cos \theta$ subtract 109 from both sides

$- 45 = -60 \cos \theta$ divide by -60

$0.75 = \cos \theta$ take the inverse cosine using a calculator

$41.41 = \theta$

Sometimes with law of cosines we have to worry about a second angle that the calculator won't give us. In this case, that would be 360 - 41.41=318.59 but that is too big an angle to be in a triangle.

Report an Error

Example Question #61 : Triangles

A triangle has side lengths a = 1 , b = 1.2 and $\angle C=150^{\circ}$

Which of the following equations can be used to find the length of side ?

Possible Answers:

$c=\frac{1.2\sin 30^{\circ}}{ \sin 150^{\circ}}$

$c=\sqrt{(1)^{2}+(1.2)^{2}-2\cdot 1\cdot 1.2\cos 150^{\circ}}$

$c=\sqrt{(1)^{2}+(1.2)^{2}-2\cdot 1\cdot 1.2\cos 30^{\circ}}$

$c=\sqrt{2\cdot 1\cdot 1.2\cos 150^{\circ}-(1)^{2}\cdot (1.2)^{2}}$

$c=\frac{1.2\sin 150^{\circ}}{\sin 30^{\circ}}$

Correct answer:

$c=\sqrt{(1)^{2}+(1.2)^{2}-2\cdot 1\cdot 1.2\cos 150^{\circ}}$

Explanation:

You are given the length of two sides of a triangle and the angle between them; therefore, you should use the Law of Cosines to find $\angle A$ , $\angle B$ , or, in this case, the length of .

$c^{2} = a^{2} + b^{2} - 2ab\cdot\cos C$

Substitute the given values for , , and $\angle C$ :

$c^{2}=(1)^{2}+(1.2)^{2}-2\cdot 1\cdot 1.2\cos 150^{\circ}$

At this point, if you are solving for , take the square root of both sides of the equation.

$\sqrt{c^{2}}=\sqrt{(1)^{2}+(1.2)^{2}-2\cdot 1\cdot 1.2\cos 150^{\circ}}$

$c=\sqrt{(1)^{2}+(1.2)^{2}-2\cdot 1\cdot 1.2\cos 150^{\circ}}$

This question merely asks for the equation, rather than the solution, so you need not simplify any further.

Two of the answer choices are equations derived from the Law of Sines. To use the Law of Sines, you must know at least one side and angle that correspond to one another, which is not the case here.

Report an Error

Example Question #21 : Law Of Cosines

Given a = 6.2 , b = 5.1 and c = 3.5 determine to the nearest degree the measure of $\angle B$ .

Possible Answers:

$34^{\circ}$

$90^{\circ}$

$56^{\circ}$

$55^{\circ}$

$35^{\circ}$

Correct answer:

$55^{\circ}$

Explanation:

We are given three sides and our desire is to find an angle, this means we must utilize the Law of Cosines. Since the angle desired is $\angle B$ the equation must be rewritten as such:

$b^{2} = a^{2} + c^{2} - 2ac\cos B$

Substituting the given values:

$5.1^{2} = 6.2^{2} + 3.5^{2} - 2\left(6.2 \right )\left(3.5 \right )\cos B$

Rearranging:

$\cos B = \frac{5.1^{2} - 3.5^{2} - 6.2^{2}}{-2\left(3.5 \right )\left(6.2)}$

Solving the right hand side and taking the inverse cosine we obtain:

$B = 55^{\circ}$

Report an Error

Example Question #21 : Law Of Cosines

If a = 9.72 , b = 13.91 and c = 12.08 , determine the measure of $\angle C$ to the nearest degree.

Possible Answers:

$93^{\circ}$

$31^{\circ}$

$103^{\circ}$

$59^{\circ}$

$58^{\circ}$

Correct answer:

$58^{\circ}$

Explanation:

This is a straightforward Law of Cosines problem since we are given three sides and desire one of the corresponding angles in the triangle. We write down the Law of Cosines to start:

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

Substituting the given values:

$12.08^{2} = 9.72^{2} + 13.91^{2} - 2\left(9.72 \right )\left(13.91 \right )\cos C$

Isolating the angle:

$\cos C = \frac{12.08^{2} - 9.72^{2} - 13.91^{2}}{-2\left(9.72 \right )\left(13.91 \right )}$

The final step is to take the inverse cosine of both sides:

$C = \cos^{-1}\left(\frac{12.08^{2} - 9.72^{2} - 13.91^{2}}{-2\left(9.72 \right )\left(13.91 \right )}\right) = 58.31^{\circ} = 58^{\circ}$

Report an Error

Example Question #311 : Trigonometry

If $\small a=5$ , $\small b=12$ , and $\small c=13$ find $\small \angle C$ to the nearest degree.

Possible Answers:

$\small 73^o$

$\small 90^{o}$

$\small 37^o$

$\small 110^o$

Correct answer:

$\small 90^{o}$

Explanation:

The problem gives the lengths of three sides and asks to find an angle. We can use the Law of Cosines to solve for the angle. Because we are solving for $\small \angle C$ , we use the equation:

$\small \small \small c^2 = a^2 + b^2 - 2abcos\angle C$

Substituting the values from the problem gives

$\small 13^2 = 5^2 + 12^2 - 2(5)(12)cos\angle C$

Isolating $\small \angle C$ by itself gives

$\small \small \angle C = cos(\frac{13^2 - 5^2 - 12^2}{(-2)(5)(12)}) = 90^o$

Report an Error

Example Question #319 : Trigonometry

If $\small a=4$ , $\small b=3$ , $\small c=5$ , find $\small \angle B$ to the nearest degree.

Possible Answers:

$\small 37^o$

$\small 100^o$

$\small 53^o$

$\small 90^o$

Correct answer:

$\small 37^o$

Explanation:

We are given the lengths of the three sides to a triangle. Therefore, we can use the Law of Cosines to find the angle being asked for. Since we are looking for $\small \angle B$ we use the equation,

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

Inputting the values we are given,

$\small 3^2 = 4^2 + 5^2 - 2(4)(5)\cos\angle B$

Next we isolate $\small \angle B$ by itself to solve for it

$\small \angle B = \cos^{-1}(\frac{3^2-4^2-5^2}{(-2)(4)(5)})$

$\small \angle B = 37^o$

Report an Error

Example Question #311 : Trigonometry

If $\small a=10$ , $\small b=6.2$ , and $\small c=4.8$ , find $\small \angle A$ to the nearest degree.

Possible Answers:

$\small 105^o$

$\small 70^o$

$\small 90^o$

$\small 130^o$

Correct answer:

$\small 130^o$

Explanation:

Because the problem provides all three sides of the triangle, we can use the Law of Cosines to solve this problem. Since we are solving for $\small \angle A$ , we use the equation

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

Substitute in the given values

$\small 10^2 = 6.2^2 + 4.8^2 - 2(6.2)(4.8)\cos\angle A$

Isolate $\small \angle A$

$\small \angle A =\cos^{-1}( \frac{10^2-6.2^2-4.8^2}{(-2)(6.2)(4.8)})$

$\small \angle A = 130^o$

Report an Error

Example Question #21 : Law Of Cosines

If $\small a=10$ , $\small b=6.7$ , $\small \angle C$ = $\small 45^o$ find $\small c$ to the nearest tenth.

Possible Answers:

$\small 8.7$

$\small 2.3$

$\small 4.0$

$\small 7.1$

Correct answer:

$\small 7.1$

Explanation:

Because we are given two sides of a triangle and the corresponding angle of the third side, we can use the Law of Cosines to find the length of side $\small c$ . To find side $\small c$ we use

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

Taking the square root of both sides gives us

$\small c = \sqrt{a^2+b^2-2ab\cos\angle C}$

Substituting in the values from the problem

$\small c = \sqrt{10^2 +6.7^2 - 2(10)(6.7)\cos45^o}$

$\small c = 7.1$

Report an Error

← Previous 1 2 3 4 Next →

View Trigonometry Tutors

Brenda
Certified Tutor

Minnesota State University-Mankato, Bachelor of Science, Health Teacher Education. Minnesota State University-Mankato, Master...

View Trigonometry Tutors

Sekou
Certified Tutor

University of New Haven, Bachelor in Arts, Mathematics.

View Trigonometry Tutors

Cynthia
Certified Tutor

North Carolina A & T State University, Bachelor of Science, Mechanical Engineering.

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

SAT Courses & Classes in Denver, LSAT Courses & Classes in Philadelphia, ACT Courses & Classes in Los Angeles, MCAT Courses & Classes in Houston, ISEE Courses & Classes in Dallas Fort Worth, GMAT Courses & Classes in Denver, GMAT Courses & Classes in San Francisco-Bay Area, GRE Courses & Classes in Philadelphia, GRE Courses & Classes in Washington DC, GRE Courses & Classes in Houston

Popular Test Prep

GRE Test Prep in Chicago, SAT Test Prep in Atlanta, GRE Test Prep in Los Angeles, ACT Test Prep in Los Angeles, SSAT Test Prep in Philadelphia, SAT Test Prep in Washington DC, GRE Test Prep in Houston, GMAT Test Prep in Atlanta, ISEE Test Prep in San Diego, ACT Test Prep in Miami

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 : Law of Cosines

Study concepts, example questions & explanations for Trigonometry

All Trigonometry Resources

Example Questions

Example Question #311 : Trigonometry

Example Question #51 : Triangles

Example Question #314 : Trigonometry

Example Question #61 : Triangles

Example Question #21 : Law Of Cosines

Example Question #21 : Law Of Cosines

Example Question #311 : Trigonometry

Example Question #319 : Trigonometry

Example Question #311 : Trigonometry

Example Question #21 : Law Of Cosines

All Trigonometry Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: