the law of cosines l.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
The Law of Cosines PowerPoint Presentation
Download Presentation
The Law of Cosines

Loading in 2 Seconds...

play fullscreen
1 / 38

The Law of Cosines - PowerPoint PPT Presentation


  • 105 Views
  • Uploaded on

The Law of Cosines. Prepared by Title V Staff: Daniel Judge, Instructor Ken Saita, Program Specialist East Los Angeles College. Click one of the buttons below or press the enter key. TOPICS. BACK. NEXT. EXIT. © 2002 East Los Angeles College. All rights reserved. Topics.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

The Law of Cosines


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
the law of cosines

The Law of Cosines

Prepared by Title V Staff:Daniel Judge, InstructorKen Saita, Program SpecialistEast Los Angeles College

Click one of the buttons below or press the enter key

TOPICS

BACK

NEXT

EXIT

© 2002 East Los Angeles College. All rights reserved.

slide2

Topics

Click on the topic that you wish to view . . .

EquationsGeneral Strategies for Using the Law of CosinesSASSSS

TOPICS

BACK

NEXT

EXIT

slide3

When solving an oblique triangle, using one of three available equations utilizing the cosine of an angle is handy. The equations are as follows:

TOPICS

BACK

NEXT

EXIT

slide4

TOPICS

BACK

NEXT

EXIT

slide5

Note:

The angle opposite a in equation 1 is .

The angle opposite b in equation 2 is .

The angle opposite c in equation 3 is .

TOPICS

BACK

NEXT

EXIT

slide7

Create an altitude h.

TOPICS

BACK

NEXT

EXIT

slide8

We’ve split our original oblique triangle into two triangles.

First Triangle

Second Triangle

TOPICS

BACK

NEXT

EXIT

slide9

First Triangle

Second Triangle

TOPICS

BACK

NEXT

EXIT

slide10

Our picture becomes:

TOPICS

BACK

NEXT

EXIT

slide11

Note the base of our triangles.

adj

adj

First Triangle

Second Triangle

TOPICS

BACK

NEXT

EXIT

slide12

Our triangles now become,

TOPICS

BACK

NEXT

EXIT

slide16

We now have,

TOPICS

BACK

NEXT

EXIT

slide17

Now, by the Pythagorean Theorem,

First Triangle

TOPICS

BACK

NEXT

EXIT

slide18

Second Triangle

TOPICS

BACK

NEXT

EXIT

slide21

The formula for the Law of Cosines makes use of three sides and the angle opposite one of those sides. We can use the Law of Cosines:

a. if we know two sides and the included angle, or

b. if we know all three sides of a triangle.

TOPICS

BACK

NEXT

EXIT

slide23

SAS

87.0°

17.0

15.0

c

From the model, we need to determine c, , and . We start by applying the law of cosines.

TOPICS

BACK

NEXT

EXIT

slide24

To solve for the missing side in this model, we use the form:

In this form,  is the angle between a and b, and c is the side opposite .

a

b

87.0°

17.0

15.0

c

TOPICS

BACK

NEXT

EXIT

slide25

Using the relationship

c2 = a2 + b2 – 2ab cos 

We get

c2 = 15.02 + 17.02 – 2(15.0)(17.0)cos 89.0°

= 225 + 289 – 510(0.0175)

= 505.10

So

c = 22.5

TOPICS

BACK

NEXT

EXIT

slide26

Now, since we know the measure of one angle and the length of the side opposite it, we can use the Law of Sines to complete the problem.

and

This gives

and

Note that due to round-off error, the angles do not add up to exactly 180°.

TOPICS

BACK

NEXT

EXIT

slide27

Three sides are known.

SSS

TOPICS

BACK

NEXT

EXIT

slide28

SSS

23.2

31.4

38.6

In this figure, we need to find the three angles, , , and .

TOPICS

BACK

NEXT

EXIT

slide29

To solve a triangle when all three sides are known we must first find one angle using the Law of Cosines. We must isolate and solve for the cosine of the angle we are seeking, then use the inverse cosine to find the angle.

TOPICS

BACK

NEXT

EXIT

slide30

We do this by rewriting the Law of Cosines equation to the following form:

In this form, the square being subtracted is the square of the side opposite the angle we are looking for.

Side to square and subtract

31.4

23.2

Angle to look for

38.6

TOPICS

BACK

NEXT

EXIT

slide31

We start by finding cos .

23.2

31.4

38.6

TOPICS

BACK

NEXT

EXIT

slide32

From the equation

we get

and

TOPICS

BACK

NEXT

EXIT

slide33

Our triangle now looks like this:

31.4

23.2

36.9°

38.6

Again, since we have the measure for both a side and the angle opposite it, we can use the Law of Sines to complete the solution of this triangle.

TOPICS

BACK

NEXT

EXIT

slide34

31.4

23.2

36.9°

38.6

Completing the solution we get the following:

and

TOPICS

BACK

NEXT

EXIT

slide35

Solving these two equations we get the following:

and

Again, because of round-off error, the angles do not add up to exactly 180.

TOPICS

BACK

NEXT

EXIT

slide36

Most of the round-off error can be avoided by storing the exact value you get for  and using that value to compute sin . Then, store sin  in your calculator’s memory also and use that value to get  and .

TOPICS

BACK

NEXT

EXIT

slide37

In this case we get the following:

If we round off at this point we get  = 36.9°,  = 54.4° and  = 88.7°.

Now the three angles add up to 180°.

TOPICS

BACK

NEXT

EXIT

slide38

End of Law of CosinesTitle V East Los Angeles College1301 Avenida Cesar ChavezMonterey Park, CA 91754Phone: (323) 265-8784Fax: (323) 415-4108Email Us At:menteprog@hotmail.comOur Website:http://www.matematicamente.org

TOPICS

BACK

NEXT

EXIT