1. Law of Sines and Law of Cosines
Geometry
Law of Sines and Law of Cosines
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2. Warm up
Use the given trigonometric ratio to determine which angle of the triangle is /A.
cos A = 15/17
sin A = 15/17
tan A = 1.875
1) 2
2) 1
3) 1
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3. Law of Sines and Law of Cosines
In this lesson, you will learn to solve any triangle. To do so, you will need to calculate trigonometric ratios for angle measures up to 180˚.
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4. Finding Trigonometric Ratios for Obtuse Angles
Use a calculator to find each trigonometric ratio. Round to the nearest hundredth.
A) sin 135˚ B) tan 98 ˚ C) cos 108 ˚
sin 135˚ = 0.71
tan 98˚= -0.72
cos 108˚=-0.31
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5. Now you try!
1) Use a calculator to find each trigonometric ratio. Round to the nearest hundredth.
a) tan 175˚ b) cos 92˚ c) sin 160˚
1a) -0.09
1b) -0.03
1c) 0.34
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6. You can use the altitude of a triangle to find a relationship between the triangle’s side lengths.
In ∆ABC, let h represent the length of the altitude from C to AB.
From the diagram, sin A = (h), and sin B = (h).
By solving for h, you find that h = bsin A
And h = a.sinB .So b sin A = a.sinB,
And (sin A/a) = (sin B/b).
You can use another altitude to show that these ratios
Equal (sin C/c). ‾ b a
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7. The Law of Sines
Theorem 1.1
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8. 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 length and a non-include angle measure (SSA).
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9. Use the Law of Sines
Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree.
Law of Sines
Substitute the given values.
Cross Products Property
Divide both sides by sin 105˚.
Next page 
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10. Use the inverse sine function to find m/s.
Law of Sines
Substitute the given
values.
Multiply both sides by 5.
Use the inverse sine function to find m/s.
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11. Now you try!
2) Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree.
2a) 2
2b) 1
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12. The Law of Sine cannot be used to solve every triangle
The Law of Sine cannot be used to solve every triangle. If you know two side lengths and the include 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.
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13. The Law of Cosines
Theorem 1.2
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14. 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).
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15. Using the Law of Cosines
Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree.
Law of Cosines
Substitute the given
values.
Simplify.
Find the square root of both sides.
Next page 
CONFIDENTIAL

16. Subtract 65 from both sides.
Law of Cosines
Substitute the given
values.
Simplify
Subtract 65 from both sides.
Solve for cos R
Use the inverse Cosine function
to find m/R.
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17. Now you try! 3) Find each measure. Round lengths to the nearest tenth
and angle measures to the nearest degree.
3a) 2
3b) 1
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18. Engineering Application
The Leaning Tower of Pisa is 56 m tall. In 1999, the tower made a 100˚ angle with the ground. To stabilize the tower, an engineer considered attaching a cable from the top of the tower to a point that is 40 m from the base. How long would the cable be, and what angle would it make with the ground? Round the length to the nearest tenth and the angle measure to the nearest degree.
Law of Cosines
Substitute the given
values.
Simplify.
Find the square root of both sides.
Next page 
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19. Substitute the Calculated values.
Law of Sines
Substitute the Calculated
values.
Multiply both sides by 56.
Use the inverse sine function to find m/A.
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20. Now you try!
4) Another engineer suggested using a cable attached from the top of the tower to a point 31 m from the base. How long would this cable be, and what angle would it make with the ground? Round the length to the nearest tenth and the angle measure to the nearest degree.
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21. Now some practice problems for you!
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22. Use a calculator to find each trigonometric ratio
Use a calculator to find each trigonometric ratio. Round to the nearest hundredth.
1) sin 100˚ ) cos 167˚ ) tan 92˚
4) tan 141˚ ) sin 150˚ ) cos 133˚
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23. Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree.7)
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24. Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree.8) 9)
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25. 10) A carpenter makes a triangular frame by joining three pieces of wood that are 20 cm, 24 cm, and 30 cm long. What are the measures of the angles of the triangle? Round to the nearest degree.
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26. Finding Trigonometric Ratios for Obtuse Angles
Let’s review
Finding Trigonometric Ratios for Obtuse Angles
Use a calculator to find each trigonometric ratio. Round to the nearest hundredth.
A sin 135˚ B tan 98 ˚ C cos 108 ˚
sin 135˚ = 0.71
tan 98˚= -0.72
cos 108˚=-0.31
CONFIDENTIAL

27. You can use the altitude of a triangle to find a relationship between the triangle’s side lengths.
In ∆ABC, let h represent the length of the altitude from C to AB.
From the diagram, sin A = (h/b), and sin B = (h/a).
By solving for h, you find that h = bsin A
And h = asin B .So b sin A = asin B,
And (sin A/a) = (sin B/b).
You can use another altitude to show that these ratios
Equal (sin C/c). ‾
CONFIDENTIAL

28. The Law of Sines
Theorem 1.1
CONFIDENTIAL

29. 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 length and a non-include angle measure (SSA).
CONFIDENTIAL

30. Use the Law of Sines
Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree.
Law of Sines
Substitute the given values.
Cross Products Property
Divide both sides by sin 105˚.
Next page 
CONFIDENTIAL

31. Use the inverse sine function to find m/s.
Law of Sines
Substitute the given
values.
Multiply both sides by 5.
Use the inverse sine function to find m/s.
CONFIDENTIAL

32. The Law of Sine cannot be used to solve every triangle
The Law of Sine cannot be used to solve every triangle. If you know two side lengths and the include 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.
CONFIDENTIAL

33. The Law of Cosines
Theorem 1.2
CONFIDENTIAL

34. 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).
CONFIDENTIAL

35. Using the Law of Cosines
Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree.
Law of Cosines
Substitute the given
values.
Simplify.
Find the square root of both sides.
Next page 
CONFIDENTIAL

36. Subtract 65 from both sides.
Law of Cosines
Substitute the given
values.
Simplify
Subtract 65 from both sides.
Solve for cos R
Use the inverse Cosine function
to find m/R.
CONFIDENTIAL

37. Engineering Application
The Leaning Tower of Pisa is 56 m tall. In 1999, the tower made a 100˚ angle with the ground. To stabilize the tower, an engineer considered attaching a cable from the top of the tower to a point that is 40 m from the base. How long would the cable be, and what angle would it make with the ground? Round the length to the nearest tenth and the angle measure to the nearest degree.
Law of Cosines
Substitute the given
values.
Simplify.
Find the square root of both sides.
Next page 
CONFIDENTIAL

38. Substitute the Calculated values.
Law of Sines
Substitute the Calculated
values.
Multiply both sides by 56.
Use the inverse sine function to find m/A.
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39. You did a great job today!
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