CK-12 Geometry - Second Edition

8.7. Extension: Laws of Sines and Cosines http://www.ck12.org

Now, use the Law of Sines to set up ratios foraandb.

sin 57◦

a

=

sin 85◦

b

=

sin 38◦

12

sin 57◦

a

=

sin 38◦

12

sin 85◦

b

=

sin 38◦

12

a·sin 38◦= 12 ·sin 57◦ b·sin 38◦= 12 ·sin 85◦

a=

12 ·sin 57◦

sin 38◦

≈ 16. 4 b=

12 ·sin 85◦

sin 38◦

≈ 19. 4

Example 2:Solve the triangle using the Law of Sines. Round decimal answers to the nearest tenth.

Solution:Set up the ratio for^6 Busing Law of Sines.

sin 95◦

27

=

sinB

16

27 ·sinB= 16 ·sin 95◦

sinB=

16 ·sin 95◦

27

→sin−^1

(

16 ·sin 95◦

27

)

= 36. 2 ◦

To findm^6 Cuse the Triangle Sum Theorem.m^6 C+ 95 ◦+ 36. 2 ◦= 180 ◦→m^6 C= 48. 8 ◦

To findc, use the Law of Sines again.sin 95
◦
27 =

sin 48. 8 ◦

c

c·sin 95◦= 27 ·sin 48. 8 ◦

c=

27 ·sin 48. 8 ◦

sin 95◦

≈ 20. 4

Law of Cosines

Law of Cosines:If 4 ABChas sides of lengtha,b, andc, thena^2 =b^2 +c^2 − 2 bccosA

b^2 =a^2 +c^2 − 2 accosB

c^2 =a^2 +b^2 − 2 abcosC

Even though there are three formulas, they are all very similar. First, notice that whatever angle is in the cosine, the
opposite side is on the other side of the equal sign.

Example 3:Solve the triangle using Law of Cosines. Round your answers to the nearest hundredth.