1. 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°

2. Law of Sines and Law of Cosines
8-5
Law of Sines and Law of Cosines
Holt Geometry

3. Example 1: Finding Trigonometric Ratios for Obtuse Angles
Use your calculator to find each trigonometric ratio. Round to the nearest hundredth.
A. tan 103°
B. cos 165°
C. sin 93°
tan 103°  –4.33
cos 165°  –0.97
sin 93°  1.00

4. 8.5 Practice (On a Separate Sheet of Paper)

5. 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).

6. 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.

7. 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.

8. 8.5 Practice (Continued)

9. 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.

10. 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

11. 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

12. Check It Out! Example 3d
Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree.
mR
PQ2 = PR2 + RQ2 – 2(PR)(RQ)cos R
Law of Cosines
Substitute the given values.
9.62 = – 2(5.9)(10.5)cos R
92.16 = – 123.9cosR
Simplify.
Subtract both sides.
–52.9 = –123.9 cosR

13. 8.5 Practice (Continued)

14. Lesson Quiz: Part I
Use a calculator to find each trigonometric ratio. Round to the nearest hundredth.
1. tan 154°
2. cos 124°
3. sin 162°
–0.49
–0.56
0.31

15. Lesson Quiz: Part II
Use ΔABC for Items 4–6. Round lengths to the nearest tenth and angle measures to the nearest degree.
4. mB = 20°, mC = 31° and b = 210. Find a.
5. a = 16, b = 10, and mC = 110°. Find c.
6. a = 20, b = 15, and c = 8.3. Find mA.
477.2
21.6
115°

16. Lesson Quiz: Part III
7. An observer in tower A sees a fire 1554 ft away at an angle of depression of 28°. To the nearest foot, how far is the fire from an observer in tower B? To the nearest degree, what is the angle of depression to the fire from tower B?
1212 ft; 37°