1. Law of Sines and Law of Cosines
8-5
Law of Sines and Law of Cosines
Warm Up
Lesson Presentation
Lesson Quiz
Holt McDougal Geometry
Holt Geometry

2. Warm Up
1. What is the third angle measure in a triangle with angles measuring 65° and 43°?
Find each value. Round trigonometric
ratios to the nearest hundredth and angle measures to the nearest degree.
2. sin 73° 3. cos 18° 4. tan 82°
5. sin-1 (0.34) 6. cos-1 (0.63) 7. tan-1 (2.75)
72°
0.96
0.95
7.12
20°
51°
70°

3. Objective
Use the Law of Sines and the Law of Cosines to solve triangles.

4. You can use the Law of Sines to solve a triangle if you are given
• two angle measures and any side length
(ASA or AAS) or
• two side lengths and a non-included angle measure (SSA).

5. Example 2A: Using the Law of Sines
Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree. FG
Law of Sines
Substitute the given values.
FG sin 39° = 40 sin 32°
Cross Products Property
Divide both sides by sin 39.

6. Example 2B: Using the Law of Sines
Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree.
mQ
Law of Sines
Substitute the given values.
Multiply both sides by 6.
Use the inverse sine function to find mQ.

7. Check It Out! Example 2a
Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree. NP
Law of Sines
Substitute the given values.
NP sin 39° = 22 sin 88°
Cross Products Property
Divide both sides by sin 39°.

8. Check It Out! Example 2b
Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree.
mL
Law of Sines
Substitute the given values.
Cross Products Property
10 sin L = 6 sin 125°
Use the inverse sine function to find mL.

9. Check It Out! Example 2d
Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree. AC
mA + mB + mC = 180°
Prop of ∆.
mA + 67° + 44° = 180°
Substitute the given values.
mA = 69°
Simplify.

10. Check It Out! Example 2D Continued
Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree.
Law of Sines
Substitute the given values.
AC sin 69° = 18 sin 67°
Cross Products Property
Divide both sides by sin 69°.

11. The Law of Sines cannot be used to solve every triangle
The Law of Sines cannot be used to solve every triangle. If you know two side lengths and the included angle measure or if you know all three side lengths, you cannot use the Law of Sines. Instead, you can apply the Law of Cosines.

12. You can use the Law of Cosines to solve a triangle if you are given
• two side lengths and the included angle measure (SAS) or
• three side lengths (SSS).

13. Example 3A: Using the Law of Cosines
Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree. XZ
XZ2 = XY2 + YZ2 – 2(XY)(YZ)cos Y
Law of Cosines
Substitute the given values.
= – 2(35)(30)cos 110°
XZ2 
Simplify.
Find the square root of both sides.
XZ  53.3

14. Example 3B: Using the Law of Cosines
Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree.
mT
RS2 = RT2 + ST2 – 2(RT)(ST)cos T
Law of Cosines
Substitute the given values.
72 = – 2(13)(11)cos T
49 = 290 – 286 cosT
Simplify.
Subtract 290 both sides.
–241 = –286 cosT

15. Example 3B Continued
Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree.
mT
–241 = –286 cosT
Solve for cosT.
Use the inverse cosine function to find mT.

16. Check It Out! Example 3a
Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree. DE
DE2 = EF2 + DF2 – 2(EF)(DF)cos F
Law of Cosines
Substitute the given values.
= – 2(18)(16)cos 21°
DE2 
Simplify.
Find the square root of both sides.
DE  6.5

17. Check It Out! Example 3b
Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree.
mK
JL2 = LK2 + KJ2 – 2(LK)(KJ)cos K
Law of Cosines
Substitute the given values.
82 = – 2(15)(10)cos K
64 = 325 – 300 cosK
Simplify.
Subtract 325 both sides.
–261 = –300 cosK

18. Check It Out! Example 3b Continued
Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree.
mK
–261 = –300 cosK
Solve for cosK.
Use the inverse cosine function to find mK.

19. Do not round your answer until the final step of the computation
Do not round your answer until the final step of the computation. If a problem has multiple steps, store the calculated answers to each part in your calculator.
Helpful Hint