the law of cosines · using the law of cosines. we must isolate and solve for the cosine of the...
TRANSCRIPT
![Page 1: The Law of Cosines · 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](https://reader034.vdocuments.mx/reader034/viewer/2022050512/5f9c9427f273ef65c02e25c3/html5/thumbnails/1.jpg)
The Law of Cosines
Prepared by Title V Staff:Daniel Judge, Instructor
Ken Saita, Program Specialist
East Los Angeles CollegeClick one of the buttons below
or press the enter key
TOPICS BACK NEXT EXIT© 2002 East Los Angeles College. All rights reserved.
![Page 2: The Law of Cosines · 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](https://reader034.vdocuments.mx/reader034/viewer/2022050512/5f9c9427f273ef65c02e25c3/html5/thumbnails/2.jpg)
TopicsClick on the topic that you wish to view . . .
EquationsGeneral Strategies for Using the Law of Cosines
SASSSS
TOPICS BACK NEXT EXIT
![Page 3: The Law of Cosines · 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](https://reader034.vdocuments.mx/reader034/viewer/2022050512/5f9c9427f273ef65c02e25c3/html5/thumbnails/3.jpg)
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
![Page 4: The Law of Cosines · 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](https://reader034.vdocuments.mx/reader034/viewer/2022050512/5f9c9427f273ef65c02e25c3/html5/thumbnails/4.jpg)
2 2 2
2 2 2
2 2 2
1) a b c 2bc cos( )2) b a c 2ac cos( )3) c a b 2ab cos( )
α
β
γ
= + −
= + −
= + −
TOPICS BACK NEXT EXIT
![Page 5: The Law of Cosines · 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](https://reader034.vdocuments.mx/reader034/viewer/2022050512/5f9c9427f273ef65c02e25c3/html5/thumbnails/5.jpg)
Note:
The angle opposite a in equation 1 is α.
The angle opposite b in equation 2 is β.
The angle opposite c in equation 3 is γ.
TOPICS BACK NEXT EXIT
![Page 6: The Law of Cosines · 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](https://reader034.vdocuments.mx/reader034/viewer/2022050512/5f9c9427f273ef65c02e25c3/html5/thumbnails/6.jpg)
Where did these three equations come from?
TOPICS BACK NEXT EXIT
![Page 7: The Law of Cosines · 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](https://reader034.vdocuments.mx/reader034/viewer/2022050512/5f9c9427f273ef65c02e25c3/html5/thumbnails/7.jpg)
Create an altitude h.
TOPICS BACK NEXT EXIT
![Page 8: The Law of Cosines · 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](https://reader034.vdocuments.mx/reader034/viewer/2022050512/5f9c9427f273ef65c02e25c3/html5/thumbnails/8.jpg)
We’ve split our original oblique triangle into two triangles.
First Triangle Second Triangle
TOPICS BACK NEXT EXIT
![Page 9: The Law of Cosines · 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](https://reader034.vdocuments.mx/reader034/viewer/2022050512/5f9c9427f273ef65c02e25c3/html5/thumbnails/9.jpg)
First Triangle Second Triangle
hsin( )a
Thus h a sin( )
γ
γ
=
= i
hsin( )c
Thus h c sin( )
α
α
=
= i
TOPICS BACK NEXT EXIT
![Page 10: The Law of Cosines · 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](https://reader034.vdocuments.mx/reader034/viewer/2022050512/5f9c9427f273ef65c02e25c3/html5/thumbnails/10.jpg)
Our picture becomes:
c sin( )αi a sin( )γi
TOPICS BACK NEXT EXIT
![Page 11: The Law of Cosines · 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](https://reader034.vdocuments.mx/reader034/viewer/2022050512/5f9c9427f273ef65c02e25c3/html5/thumbnails/11.jpg)
Note the base of our triangles.
adj adj
c sin( )αi a sin( )γi
First Triangle Second Triangleadjcos( )c
adj c cos( )
α
α
=
= i
adjcos( )a
adj a cos( )
γ
γ
=
= iTOPICS BACK NEXT EXIT
![Page 12: The Law of Cosines · 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](https://reader034.vdocuments.mx/reader034/viewer/2022050512/5f9c9427f273ef65c02e25c3/html5/thumbnails/12.jpg)
Our triangles now become,
c sin( )αi a sin( )γi
a cos( )γic cos( )αi
TOPICS BACK NEXT EXIT
![Page 13: The Law of Cosines · 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](https://reader034.vdocuments.mx/reader034/viewer/2022050512/5f9c9427f273ef65c02e25c3/html5/thumbnails/13.jpg)
*Consider two important relationships:
1) c sin( ) a sin( )2) c cos( ) a cos( ) b
α γα γ=+ =
i ii i
TOPICS BACK NEXT EXIT
![Page 14: The Law of Cosines · 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](https://reader034.vdocuments.mx/reader034/viewer/2022050512/5f9c9427f273ef65c02e25c3/html5/thumbnails/14.jpg)
Using Relationship 1, we obtain:
c sin( )αia sin( )γi
a cos( )γic cos( )αi
TOPICS BACK NEXT EXIT
![Page 15: The Law of Cosines · 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](https://reader034.vdocuments.mx/reader034/viewer/2022050512/5f9c9427f273ef65c02e25c3/html5/thumbnails/15.jpg)
Take a closer look at Relationship 2.
c cos( ) a cos( ) bα γ+ =i i
a cos( ) b c cos( )γ α= −i ic cos( ) b a cos( )α γ= −i i
TOPICS BACK NEXT EXIT
![Page 16: The Law of Cosines · 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](https://reader034.vdocuments.mx/reader034/viewer/2022050512/5f9c9427f273ef65c02e25c3/html5/thumbnails/16.jpg)
We now have,
c sin( )αia sin( )γi
b c cos( )α− ib a cos( )γ− i
TOPICS BACK NEXT EXIT
![Page 17: The Law of Cosines · 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](https://reader034.vdocuments.mx/reader034/viewer/2022050512/5f9c9427f273ef65c02e25c3/html5/thumbnails/17.jpg)
Now, by the Pythagorean Theorem,
First Triangle
2 22
2 2 2 2 2
2 2 2 2 2
2 2 2 2
2 2
2 2 2
sin ( ) cos ( ) 1Thus c a b 2ab c
c a sin( ) b a cos( ) a sin ( ) b ab cos( ) ab cos( ) a cos ( ) a sin ( ) a cos ( ) b 2ab cos( ) a sin ( ) cos ( ) b 2ab cos( )
But γ γ
γ γ
γ γ γ γγ γ γ
γ γ γ
+ =
= + −
= + −
= + − − +
= + + −
= + + −
os( )γ
TOPICS BACK NEXT EXIT
![Page 18: The Law of Cosines · 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](https://reader034.vdocuments.mx/reader034/viewer/2022050512/5f9c9427f273ef65c02e25c3/html5/thumbnails/18.jpg)
Second Triangle
[ ] [ ]2 22
2 2 2 2 2
2 2 2 2 2
2 2 2 2
2 2
2 2 2
a c sin( ) b c cos( )
c sin ( ) b bc cos( ) bc cos( ) c cos ( ) c sin ( ) c cos ( ) b 2bc cos( ) c sin ( ) cos ( ) b 2bc cos( )
also, sin ( ) cos ( ) 1thus a b c 2bc cos( )
α α
α α α α
α α α
α α α
α α
α
= + −
= + − − +
= + + −
= + + − + =
= + −
TOPICS BACK NEXT EXIT
![Page 19: The Law of Cosines · 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](https://reader034.vdocuments.mx/reader034/viewer/2022050512/5f9c9427f273ef65c02e25c3/html5/thumbnails/19.jpg)
Why don’t you try the third equation.
TOPICS BACK NEXT EXIT
![Page 20: The Law of Cosines · 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](https://reader034.vdocuments.mx/reader034/viewer/2022050512/5f9c9427f273ef65c02e25c3/html5/thumbnails/20.jpg)
General Strategies for Usingthe Law of Cosines
TOPICS BACK NEXT EXIT
![Page 21: The Law of Cosines · 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](https://reader034.vdocuments.mx/reader034/viewer/2022050512/5f9c9427f273ef65c02e25c3/html5/thumbnails/21.jpg)
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
![Page 22: The Law of Cosines · 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](https://reader034.vdocuments.mx/reader034/viewer/2022050512/5f9c9427f273ef65c02e25c3/html5/thumbnails/22.jpg)
Two sides and one angles are known.
SAS
TOPICS BACK NEXT EXIT
![Page 23: The Law of Cosines · 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](https://reader034.vdocuments.mx/reader034/viewer/2022050512/5f9c9427f273ef65c02e25c3/html5/thumbnails/23.jpg)
SAS
87.0°17.0
α
15.0
β
c
From the model, we need to determine c, α, and β. We start by applying the law of cosines.
TOPICS BACK NEXT EXIT
![Page 24: The Law of Cosines · 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](https://reader034.vdocuments.mx/reader034/viewer/2022050512/5f9c9427f273ef65c02e25c3/html5/thumbnails/24.jpg)
To solve for the missing side in this model, we use the form:
2 2 2 2 cosc a b ab γ= + −In this form, γ is the angle between aand b, and c is the side opposite γ.
87.0°15.017.0
cβα
γ ab
TOPICS BACK NEXT EXIT
![Page 25: The Law of Cosines · 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](https://reader034.vdocuments.mx/reader034/viewer/2022050512/5f9c9427f273ef65c02e25c3/html5/thumbnails/25.jpg)
Using the relationship
c2 = a2 + b2 – 2ab cos γ
We get
c2 = 15.02 + 17.02 – 2(15.0)(17.0)cos 89.0°
= 225 + 289 – 510(0.0175)
= 505.10So c = 22.5
TOPICS BACK NEXT EXIT
![Page 26: The Law of Cosines · 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](https://reader034.vdocuments.mx/reader034/viewer/2022050512/5f9c9427f273ef65c02e25c3/html5/thumbnails/26.jpg)
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.
sin 87.0 sin22.1 15.0
α=
sin 87.0 sin22.1 17.0
β=and
This gives42.7α = and 50.2β =
Note that due to round-off error, the angles do not add up to exactly 180°.
TOPICS BACK NEXT EXIT
![Page 27: The Law of Cosines · 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](https://reader034.vdocuments.mx/reader034/viewer/2022050512/5f9c9427f273ef65c02e25c3/html5/thumbnails/27.jpg)
Three sides are known.
SSS
TOPICS BACK NEXT EXIT
![Page 28: The Law of Cosines · 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](https://reader034.vdocuments.mx/reader034/viewer/2022050512/5f9c9427f273ef65c02e25c3/html5/thumbnails/28.jpg)
SSS
31.4 23.2
α β
γ
38.6
In this figure, we need to find the three angles, α, β, and γ.
TOPICS BACK NEXT EXIT
![Page 29: The Law of Cosines · 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](https://reader034.vdocuments.mx/reader034/viewer/2022050512/5f9c9427f273ef65c02e25c3/html5/thumbnails/29.jpg)
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
![Page 30: The Law of Cosines · 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](https://reader034.vdocuments.mx/reader034/viewer/2022050512/5f9c9427f273ef65c02e25c3/html5/thumbnails/30.jpg)
We do this by rewriting the Law of Cosines equation to the following form:
2 2 2
cos2
b c abc
α + −=
In this form, the square being subtracted is the square of the side opposite the angle we are looking for.
Side to square and subtract31.4 23.2
α β
γ
Angle to look for
38.6
TOPICS BACK NEXT EXIT
![Page 31: The Law of Cosines · 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](https://reader034.vdocuments.mx/reader034/viewer/2022050512/5f9c9427f273ef65c02e25c3/html5/thumbnails/31.jpg)
We start by finding cos α.
31.4 23.2
α β
γ
38.6
2 2 231.4 38.6 23.2cos2(31.4)(38.6)
α + −=
TOPICS BACK NEXT EXIT
![Page 32: The Law of Cosines · 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](https://reader034.vdocuments.mx/reader034/viewer/2022050512/5f9c9427f273ef65c02e25c3/html5/thumbnails/32.jpg)
From the equation2 2 231.4 38.6 23.2cos
2(31.4)(38.6)α + −=
we getcos 0.7993α =
and36.9α =
TOPICS BACK NEXT EXIT
![Page 33: The Law of Cosines · 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](https://reader034.vdocuments.mx/reader034/viewer/2022050512/5f9c9427f273ef65c02e25c3/html5/thumbnails/33.jpg)
Our triangle now looks like this:
31.4 23.2
β
γ
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
![Page 34: The Law of Cosines · 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](https://reader034.vdocuments.mx/reader034/viewer/2022050512/5f9c9427f273ef65c02e25c3/html5/thumbnails/34.jpg)
31.4 23.2
β
γ
36.9°38.6
Completing the solution we get the following:sin sin 36.931.4 23.2
β= and
sin sin 36.938.6 23.2
γ=
TOPICS BACK NEXT EXIT
![Page 35: The Law of Cosines · 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](https://reader034.vdocuments.mx/reader034/viewer/2022050512/5f9c9427f273ef65c02e25c3/html5/thumbnails/35.jpg)
Solving these two equations we get the following:
sin 0.9990γ =sin 0.8126β =and
54.4β = 87.3γ =
Again, because of round-off error, the angles do not add up to exactly 180°.
TOPICS BACK NEXT EXIT
![Page 36: The Law of Cosines · 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](https://reader034.vdocuments.mx/reader034/viewer/2022050512/5f9c9427f273ef65c02e25c3/html5/thumbnails/36.jpg)
Most of the round-off error can be avoided by storing the exact value you get for α and using that value to compute sin α.
Then, store sin α in your calculator’s memory also and use that value to get β and γ.
TOPICS BACK NEXT EXIT
![Page 37: The Law of Cosines · 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](https://reader034.vdocuments.mx/reader034/viewer/2022050512/5f9c9427f273ef65c02e25c3/html5/thumbnails/37.jpg)
In this case we get the following:
cos .799346556236.9322517
sin .600870271454.4143797288.65337196
α
αα
βγ
=
==
==
If we round off at this point we get α = 36.9°, β = 54.4° and γ = 88.7°.
Now the three angles add up to 180°.
TOPICS BACK NEXT EXIT
![Page 38: The Law of Cosines · 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](https://reader034.vdocuments.mx/reader034/viewer/2022050512/5f9c9427f273ef65c02e25c3/html5/thumbnails/38.jpg)
End of Law of Cosines
Title V East Los Angeles College
1301 Avenida Cesar ChavezMonterey Park, CA 91754
Phone: (323) 265-8784Fax: (323) 415-4108
Email Us At:[email protected]
Our Websites:http://www.matematicamente.org
http://mente.dhs.org
TOPICS BACK NEXT EXIT