1. The Law of Sines & Cosines
Keeper 15
Pre-Calculus

2. Law of Sines The Law of Sines In any triangle ABC, B a c C A b
The Law of Sines applies to AAS, ASA, SSA(special case).
The Law of Sines
In any triangle ABC, A B C a b c
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3. Example Solution: Draw the triangle. We have AAS.
In , e = 4.56, E = 43º, and G = 57º. Solve the triangle.
Solution:
Draw the triangle.
We have AAS.
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4. Example Find F: F = 180º – (43º + 57º) = 80º
Solution continued
Find F: F = 180º – (43º + 57º) = 80º
Use law of sines to find the other two sides.
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5. Example We have solved the triangle. Solution continued
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6. Check for Understanding.
Fill-In F
iRespond Question
A.) 17.2;;
B.)
C.)
D.)
E.)
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8. Check for Understanding
Fill-In
iRespond Question .
A.) 21;;
B.)
C.)
D.)
E.)
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12. Determine if the data supports 1 unique triangle, 2 triangles
That are not congruent or 0 triangles. F
Fill-In
iRespond Question
A.) 1;;
B.)
C.)
D.)
E.)
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13. 25
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14. Determine if the data supports 1 unique triangle, 2 triangles
That are not congruent or 0 triangles. F
Fill-In
iRespond Question
A.) 2;;
B.)
C.)
D.)
E.)
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16. Determine if the data supports 1 unique triangle, 2 triangles
That are not congruent or 0 triangles. F
Fill-In
iRespond Question
A.) 0;;
B.)
C.)
D.)
E.)
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17. Copyright © 2009 Pearson Education, Inc.

18. Law of Cosines The Law of Cosines In any triangle ABC, B a c C A b
Thus, in any triangle, the square of a side is the sum of the squares of the other two sides, minus twice the product of those sides and the cosine of the included angle. When the included angle is 90º, the law of cosines reduces to the Pythagorean theorem.
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19. When to use the Law of Cosines
The Law of Cosines is used to solve triangles given two sides and the included angle (SAS) or given three sides (SSS).
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20. Example Solution: Draw and label a triangle.
In !ABC, a = 32, c = 48, and B = 125.2º. Solve the triangle.
Solution:
Draw and label a triangle.
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21. Example Use the law of cosines to find the third side, b.
Solution continued
Use the law of cosines to find the third side, b.
We need to find the other two angle measures. We can use either the law of sines or law of cosines. Using the law of cosines avoids the possibility of the ambiguous case. So use the law of cosines.
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22. Example Find angle A. Now find angle C. C ≈ 180º – (125.2º + 22º)
Solution continued
Find angle A.
Now find angle C.
C ≈ 180º – (125.2º + 22º)
C ≈ 32.8º
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23. Example Solution: Draw and label a triangle.
Solve !RST, r = 3.5, s = 4.7, and t = 2.8.
Solution:
Draw and label a triangle.
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24. Example Similarly, find angle R. Solution continued
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25. Example Now find angle T. T ≈ 180º – (95.86º + 47.80º) ≈ 36.34º
Solution continued
Now find angle T.
T ≈ 180º – (95.86º º) ≈ 36.34º
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26. The Area of a Triangle
The area of any is one half the product of he lengths of two sides and the sine of the included angle:
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27. Example
A university landscaping architecture department is designing a garden for a triangular area in a dormitory complex. Two sides of the garden, formed by the sidewalks in front of buildings A and B, measure 172 ft
and 186 ft, respectively, and together form a 53º angle. The third side of the garden,formed by the sidewalk along Crossroads Avenue, measures 160 ft. What is the area of the garden to the nearest square foot?
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28. Example The area of the garden is approximately 12,775 ft2.
Solution: Use the area formula.
The area of the garden is approximately 12,775 ft2.
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29. Note: s is the “semi-perimeter.”