1. Splash Screen

2. The top of a signal tower is 250 feet above sea level
The top of a signal tower is 250 feet above sea level. The angle of depression from the top of the tower to a passing ship is 19°. How far is the foot of the tower from the ship?
A. about 81.4 ft
B. about ft
C. about 726 ft
D. about 804 ft
5-Minute Check 4

3. You used trigonometric ratios to solve right triangles.
Use the Law of Sines to solve triangles.
Use the Law of Cosines to solve triangles.
Then/Now

4. Law of Sines
Law of Cosines
Vocabulary

5. Concept

6. Find p. Round to the nearest tenth.
Law of Sines (AAS or ASA)
Find p. Round to the nearest tenth.
We are given measures of two angles and a nonincluded side, so use the Law of Sines to write a proportion.
Example 1

7. Cross Products Property
Law of Sines (AAS or ASA)
Law of Sines
Cross Products Property
Divide each side by sin
Use a calculator.
Answer: p ≈ 4.8
Example 1

8. Find c to the nearest tenth.
B. 29.9
C. 7.8
D. 8.5
Example 1

9. Find x. Round to the nearest tenth.
Law of Sines (ASA)
Find x. Round to the nearest tenth. 6 x
57°
Example 2

10. 6 sin 50 = x sin 73 Cross Products Property
Law of Sines (ASA)
Law of Sines
mB = 50, mC = 73, c = 6
6 sin 50 = x sin 73 Cross Products Property
Divide each side by sin 73.
4.8 = x Use a calculator.
Answer: x ≈ 4.8
Example 2

11. Find x. Round to the nearest degree.
B. 10
C. 12
D. 14
43° x
Example 2

12. Concept

13. Find x. Round to the nearest tenth.
Law of Cosines (SAS)
Find x. Round to the nearest tenth.
Use the Law of Cosines since the measures of two sides and the included angle are known.
Example 3

14. Take the square root of each side.
Law of Cosines (SAS)
Law of Cosines
Simplify.
Take the square root of each side.
Use a calculator.
Answer: x ≈ 18.9
Example 3

15. Find r if s = 15, t = 32, and mR = 40. Round to the nearest tenth.
B. 44.5
C. 22.7
D. 21.1
Example 3

16. Find mL. Round to the nearest degree.
Law of Cosines (SSS)
Find mL. Round to the nearest degree.
Law of Cosines
Simplify.
Example 4

17. Subtract 754 from each side.
Law of Cosines (SSS)
Subtract 754 from each side.
Divide each side by –270.
Solve for L.
Use a calculator.
Answer: mL ≈ 49
Example 4

18. Find mP. Round to the nearest degree.
B. 51°
C. 56°
D. 69°
Example 4

19. Indirect Measurement
AIRCRAFT From the diagram of the plane shown, determine the approximate width of each wing. Round to the nearest tenth meter.
Example 5

20. Use the Law of Sines to find KJ.
Indirect Measurement
Use the Law of Sines to find KJ.
Law of Sines
Cross products
Example 5

21. Answer: The width of each wing is about 16.9 meters.
Indirect Measurement
Divide each side by sin
Simplify.
Answer: The width of each wing is about 16.9 meters.
Example 5

22. The rear side window of a station wagon has the shape shown in the figure. Find the perimeter of the window if the length of DB is 31 inches. Round to the nearest tenth.
A in.
B in.
C in.
D in.
Example 5

23. Solve triangle PQR. Round to the nearest degree.
Solve a Triangle
Solve triangle PQR. Round to the nearest degree.
Since the measures of three sides are given (SSS), use the Law of Cosines to find mP.
p2 = r2 + q2 – 2pq cos P Law of Cosines
82 = – 2(9)(7) cos P p = 8, r = 9, and q = 7
Example 6

24. –66 = –126 cos P Subtract 130 from each side.
Solve a Triangle
64 = 130 – 126 cos P Simplify.
–66 = –126 cos P Subtract 130 from each side.
Divide each side by –126.
Use the inverse cosine ratio.
Use a calculator.
Example 6

25. Use the Law of Sines to find mQ.
Solve a Triangle
Use the Law of Sines to find mQ.
Law of Sines
mP ≈ 58, p = 8, q = 7
Multiply each side by 7.
Use the inverse sine ratio.
Use a calculator.
Example 6

26. By the Triangle Angle Sum Theorem, mR ≈ 180 – (58 + 48) or 74.
Solve a Triangle
By the Triangle Angle Sum Theorem, mR ≈ 180 – ( ) or 74.
Answer: Therefore, mP ≈ 58; mQ ≈ 48 and mR ≈ 74.
Example 6

27. Solve ΔRST. Round to the nearest degree.
A. mR = 82, mS = 58, mT = 40
B. mR = 58, mS = 82, mT = 40
C. mR = 82, mS = 40, mT = 58
D. mR = 40, mS = 58, mT = 82
Example 6

28. Concept

29. End of the Lesson