1. The Law of Sines and the Law of Cosines
LESSON 8–6
The Law of Sines and the Law of Cosines

2. Five-Minute Check (over Lesson 8–5) TEKS Then/Now New Vocabulary
Theorem 8.10: Law of Sines
Example 1: Law of Sines (AAS)
Example 2: Law of Sines (ASA)
Theorem 8.11: Law of Cosines
Example 3: Law of Cosines (SAS)
Example 4: Law of Cosines (SSS)
Example 5: Real-World Example: Indirect Measurement
Example 6: Solve a Triangle
Concept Summary: Solving a Triangle
Lesson Menu

3. Name the angle of depression in the figure.
A. URT
B. SRT
C. RST
D. SRU
5-Minute Check 1

4. Find the angle of elevation of the Sun when a 6-meter flagpole casts a 17-meter shadow.
A. about 70.6°
B. about 60.4°
C. about 29.6°
D. about 19.4°
5-Minute Check 2

5. After flying at an altitude of 575 meters, a helicopter starts to descend when its ground distance from the landing pad is 13.5 kilometers. What is the angle of depression for this part of the flight?
A. about 1.8°
B. about 2.4°
C. about 82.4°
D. about 88.6°
5-Minute Check 3

6. 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

7. Jay is standing 50 feet away from the Eiffel Tower and measures the angle of elevation to the top of the tower as 87.3°. Approximately how tall is the Eiffel Tower?
A. 50 ft
B. 104 ft
C ft
D ft
5-Minute Check 5

8. Mathematical Processes G.1(E), G.1(F)
Targeted TEKS
G.9(A) Determine the lengths of sides and measures of angles in a right triangle by applying the trigonometric ratios
sine, cosine, and tangent to solve problems.
Mathematical Processes
G.1(E), G.1(F)
TEKS

9. 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

10. Law of Sines
Law of Cosines
Vocabulary

11. Concept

12. Find p. Round to the nearest tenth.
Law of Sines (AAS)
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

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

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

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

16. 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

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

18. Concept

19. 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

20. 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

21. 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

22. 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

23. 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

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

25. 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

26. 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

27. 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

28. 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

29. 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

30. –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

31. 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

32. 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

33. 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

34. Concept

35. The Law of Sines and the Law of Cosines
LESSON 8–6
The Law of Sines and the Law of Cosines