The Law of Cosines

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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.

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## The Law of Cosines

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### 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

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Topics

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EquationsGeneral Strategies for Using the Law of CosinesSASSSS

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

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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 .

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Create an altitude h.

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First Triangle

Second Triangle

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First Triangle

Second Triangle

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Our picture becomes:

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Note the base of our triangles.

First Triangle

Second Triangle

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Our triangles now become,

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We now have,

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Now, by the Pythagorean Theorem,

First Triangle

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Second Triangle

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### General Strategies for Usingthe Law of Cosines

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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.

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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.

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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

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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

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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°.

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Three sides are known.

SSS

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SSS

23.2

31.4

38.6

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

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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.

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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

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We start by finding cos .

23.2

31.4

38.6

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From the equation

we get

and

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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.

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31.4

23.2

36.9°

38.6

Completing the solution we get the following:

and

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Solving these two equations we get the following:

and

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

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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 .

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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°.

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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

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