Solve Triangles (SAS): Solve for an Unknown Side

[Pages:2]Objective: Apply the Law of Cosines: SAS

Solve Triangles (SAS): Solve for an Unknown Side

When you know two sides and an included angle of a triangle (SAS), you can use the Law of Cosines to solve for the other side. Consider the triangle ABC with the measures in the table.

C

b

a

A=35? B=

C=

a=

b=50

c=69

A

B

c

Since we know angle A we use a2 = b2 + c2 - 2bc cos A from the Law of Cosines

a2 = b2 + c2 - 2bc cos A a = b2 + c2 - 2bc cos A a = 502 + 692 - 2(50)(69) cos 35 a 40.1104836

Now use the Law of Sines to solve for B.

sin B = sin A

b

a

sin B = sin 35 50 40.1104836

B = sin-150 sin 35 45.64o 40.1104836

Note that we store that value in "x" for our next computation.

And C 99.36o

A=35? B45.64? C99.36?

a40.11 b=50

c=69

Objective: Apply the Law of Cosines: SAS

Solve Triangles (SSS): Solve for an Unknown Angle

When you know three sides of a triangle (SSS), you can use the Law of Cosines to solve for an angle. Consider the triangles ABC with the measures in the table.

C

b

a

A

B

c

A=

a=40

B=

b=50

C=

c=70

We can solve for angle A using a2 = b2 + c2 - 2bc cos A

a2 = b2 + c2 - 2bc cos A

cos A = b2 + c2 - a2 2bc

A

=

cos-1

b2

+ c2 - 2bc

a2

A

=

cos-1

502 + 702 - 402 2(50)(70)

A 34.05

Now we could use the Law of Sines or Cosines to solve for B. We will use b2 = a2 + c2 - 2ac cos B

cos B = a2 + c2 - b2 2ac

B

=

cos-1

a2

+ c2 - 2ac

b2

B

=

cos-1

402 + 702 - 502 2(40)(70)

B 44.42

Note that we store that value in "A" for a later computation.

And C 101.54o

A34.05?

a=40

B44.42?

b=50

C101.54 ?

c=70

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