CK-12 Geometry Concepts

http://www.ck12.org Chapter 8. Right Triangle Trigonometry

• Any two angles and one side.


• Two sides and the non-included angle.

Law of Cosines:If 4 ABChas sides of lengtha,b, andc, then:

a^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.

UseLawofCosineswhengiven:

• Two sides and the included angle.


• All three sides.

Example A

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

First, to findm^6 A, we can use the Triangle Sum Theorem.

m^6 A+ 85 ◦+ 38 ◦= 180 ◦

m^6 A= 57 ◦

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 B

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

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 ◦