aim: law of cosines course: alg. 2 & trig. aim: what is the law of cosines? do now: if the...

Aim: Law of Cosines Course: Alg. 2 & Trig.

Aim: What is the Law of Cosines?

Do Now:

If the measures of two sides and theincluded angle of a triangle are known,can the size (area) and shape of the triangle be determine?

Yes


SAS SASSAS SAS

B

A

C

2 sides and included angle:

AB A’B’, BC B’C’, B B’

2 sides and included angle:

AB A’B’, BC B’C’, B B’

B’

A’

C’

B’

A’

C’

Shortcut for proving congruence in triangles:

Measurements showed: ABC A’B’C’ABC A’B’C’

SAS


Area of Triangle

y

x

(b cos A, b sin A)

b

c

a h

A B

C

Area of ABC 1

2ab sinC

1

2ac sinB

1

2bc sin A

The area of a triangle is equal to one-halfthe product of the measures of two sidesand the sine of the angle between them.

A case


Area of Triangle & Law of Cosines

y

x

(b cos A, b sin A)

b

c

a

A B(c,0)

C

Use the distance formula to find length of a

(0,0)

212

212 )()( yyxxd

a (bcos A c)2 (bsinA 0)2

a (bcos A c)2 (bsinA 0)2

a (b2 cos2 A 2bc cos A c 2) (b2 sin2 A)

a2 b2 cos2 A 2bc cos A c2 b2 sin2 A

a2 b2(cos 2 A sin2 A) 2bc cosA c2

a2 b2 2bc cos A c2

a2 b2 c 2 2bc cos A

AB = cAC = bBC = a


Law of Cosines

a2 b2 c 2 2bc cos A

The square of the measure of one side of atriangle is equal to the sum of the squaresof the measures of the other two sides minustwice the product of the measures of thesetwo sides and the cosine of the angle betweenthem.

a2 b2 c 2 2bc cos A

b2 a2 c 2 2ac cosB

c 2 a2 b2 2abcosC

Law of Cosines


What does it all mean?

a2 b2 c 2 2bc cos A

If you do know the measures of two sides and the included angle of a triangle, thenyou can determine the size and shape of the triangle.

a = ?

A B

C

35

b

c

If I know the measuresof A, b, and c, thenI can find the measureof side a.

ex. Find a if mA = 35, b = 16, and c = 7

a2 162 72 2(16)(7)cos35

a2 121.5099421to nearest tenth

16

7

a 11.0

11


Model Problem

b2 a2 c 2 2ac cosB

In ∆ABC, if a = 4, c = 6, and cos B = 1/16,find b.

b2 42 62 2(4)(6)(1/ 16)

Appropriateversion of Lawof Cosines

Substituteand solve:

b2 49

b 7 must be positive

B

A

Ca

bc


Draw:

Model Problem

t 2 r 2 s 2 2rscosT

In ∆RST, if r = 11, s = 12, and mT = 120,find t to the nearest integer.

Appropriateversion of Lawof Cosines interms of r, s &T


t 2 265 264( .5)

S

RT

t

s = 12

r = 11120º

t 2 112 122 2(11)(12) cos120

t2 = 397

t = 19.92485885must be positive

t = 20 to nearestinteger


Finding Angle Measures

In ∆ABC, a = 5, b = 7, and c = 10. Find cos B.

Appropriateversion of Lawof Cosines solvedfor cos B


b2 a2 c 2 2ac cosB

cosB a2 c2 b2

2ac

cosB 52 102 72

2(5)(10)

cosB .76

cosB a2 c2 b2

2ac

cos A b2 c2 a2

2bc

cosC a2 b2 c 2

2ab

Law of Cosinessolved for cosine of angles


Model Problem

In isosceles triangle RED, RE = ED = 5 andRD = 8. Find the measure of the vertex angle,E, to the nearest degree.

cosE r 2 d2 32

2rd

E

DR

t r = 5d = 5

120ºe = 8Draw:

Appropriateversion of Lawof Cosines solvedfor cos E

cosE 52 52 82

2(5)(5)

cosE 52 52 82

2(5)(5)

cos E = -.28

Substitute &solve:

mE ≈ 106


Regents Question

In triangle DEF, side e = 10, f = 8 and mD = 110. Find the length of the third side to the nearest tenth.

1) 218.7 2) 109.3 3) 14.8 4) 10.5


The Product Rule

A surveyor at point P sights two points X and Y that are on opposite sides of a lake. If P is 200 m. from X and 350 m. from Y, and mXPY = 40, find the distance from X to Y to the nearest meter.


Model Problem

Some nylon fabric will be cut to cover thekite frame shown below. Diagonal AC is29 inches. What size should the angles beat A, B, C, and D?

B

D

C

A 16 in.

26 in.

16 in.

26 in.

aim: law of cosines course: alg. 2 & trig. aim: what is the law of cosines? do now: if the...

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