1. EQ: What is the law of sines, and how can we use it to solve right triangles?

2. EQ: What is the law of sines, and how can we use it to solve right triangles?
The Law of Sines allows you to solve a triangle as long as you know either of the following:
1. Two angle measures and any side length–angle-angle-side (AAS) or angle-side-angle (ASA) information
2. Two side lengths and the measure of an angle that is not between them–side-side-angle (SSA) information

3. Using the Law of Sines for AAS and ASA
EQ: What is the law of sines, and how can we use it to solve right triangles?
Using the Law of Sines for AAS and ASA
Solve the triangle. Round to the nearest tenth.
Step 1. Find the third angle measure.
mD + mE + mF = 180°
Triangle Sum Theorem.
Substitute 33° for mD and 28° for mF.
33° + mE + 28° = 180°
mE = 119°
Solve for mE.

4. Step 2 Find the unknown side lengths.
EQ: What is the law of sines, and how can we use it to solve right triangles?
Step 2 Find the unknown side lengths.
sin D
sin F d f =
sin E
sin F e f =
Law of Sines.
sin 33°
sin 28° d 15 =
sin 119°
sin 28° e 15 =
Substitute.
Cross
multiply.
d sin 28° = 15 sin 33°
e sin 28° = 15 sin 119°
e =
15 sin 119°
sin 28°
e ≈ 27.9
d =
15 sin 33°
sin 28°
d ≈ 17.4
Solve for the
unknown side.

5. Using the Law of Sines for AAS and ASA
EQ: What is the law of sines, and how can we use it to solve right triangles?
Using the Law of Sines for AAS and ASA Q r
Solve the triangle. Round to the nearest tenth.
Step 1 Find the third angle measure.
Triangle Sum Theorem
mP = 180° – 36° – 39° = 105°

6. Solve the triangle. Round to the nearest tenth.
EQ: What is the law of sines, and how can we use it to solve right triangles? Q r
Solve the triangle. Round to the nearest tenth.
Step 2 Find the unknown side lengths.
sin P
sin Q p q =
Law of Sines.
sin P
sin R p r =
sin 105°
sin 36° 10 q =
sin 105°
sin 39° 10 r =
Substitute.
q =
10 sin 36°
sin 105°
≈ 6.1
r =
10 sin 39°
≈ 6.5

7. Solve the triangle. Round to the nearest tenth.
EQ: What is the law of sines, and how can we use it to solve right triangles?
Solve the triangle. Round to the nearest tenth.
Step 1 Find the third angle measure.
mH + mJ + mK = 180°
Substitute 42° for mH and 107° for mJ.
42° + 107° + mK = 180°
mK = 31°
Solve for mK.

8. Step 2 Find the unknown side lengths.
EQ: What is the law of sines, and how can we use it to solve right triangles?
Step 2 Find the unknown side lengths.
sin H
sin J h j =
sin K
sin H k h =
Law of Sines.
sin 42°
sin 107° h 12 =
sin 31°
sin 42° k
8.4 =
Substitute.
Cross
multiply.
h sin 107° = 12 sin 42°
8.4 sin 31° = k sin 42°
k =
8.4 sin 31°
sin 42°
k ≈ 6.5
h =
12 sin 42°
sin 107°
h ≈ 8.4
Solve for the
unknown side.

9. Solve the triangle. Round to the nearest tenth.
EQ: What is the law of sines, and how can we use it to solve right triangles?
Solve the triangle. Round to the nearest tenth.
Step 1 Find the third angle measure.
Triangle Sum Theorem
mN = 180° – 56° – 106° = 18°

10. Solve the triangle. Round to the nearest tenth.
EQ: What is the law of sines, and how can we use it to solve right triangles?
Solve the triangle. Round to the nearest tenth.
Step 2 Find the unknown side lengths.
sin N
sin M n m =
Law of Sines.
sin M
sin P m p =
sin 18°
sin 106°
1.5 m =
sin 106°
sin 56°
4.7 p =
Substitute.
m =
1.5 sin 106°
sin 18°
≈ 4.7
p =
4.7 sin 56°
sin 106°
≈ 4.0