Quiz 13.5 Solve for the missing angle and sides of Triangle ABC where B = 25º, b = 15, C = 107º Triangle ABC where B = 25º, b = 15, C = 107º 1. A = ? 2.

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Quiz 13.5 Solve for the missing angle and sides of Triangle ABC where B = 25º, b = 15, C = 107º Triangle ABC where B = 25º, b = 15, C = 107º 1. A = ? 2. a = ? 3. c = ? 4. Following triangle is one of ambiguous cases. Are there 0, 1, or 2 triangles for this case? Are there 0, 1, or 2 triangles for this case? A = 36.9º, b = 5, a = 4

13.5 The Law of Cosines

What you’ll learn about Solving Triangles (SAS, SSS) Triangle Area Applications … and why The Law of Cosines is an important extension of the Pythagorean theorem, with many applications.

Law of Cosines

SSS Case B C 5 7A9 1. Draw the Triangle! a = 9, b = 5, c = 7 2.Solve for the largest angle. Write the formula. Write the formula. 3. Plug numbers into the formula. 4. Solve for the unknown value in the formula. 95.7º The angle is the only unknown in the equation. unknown in the equation.

SSS Case C B 5 7A9 5.Use Law of Sines to solve the remainder to solve the remainder of the triangle. of the triangle. a = 9, b = 7, c = º B = 50.7º C = 180 – 50.7 – 95.7 C = “SSS = Second step sines”

Solving a Triangle (SSS) Your turn: 1.Draw and label the triangle. 2.Solve for the biggest angle first (C). 3.Use Law a Sines to solve for angle B. 4.A = ?

SAS CaseCB a 5 A Draw the Triangle! A = 20º, b = 5, c = 11 2.Solve for the missing side. Write the formula. Write the formula. 3. Plug numbers into the formula. 4. Solve for the unknown value in the formula. 20º

SAS CaseCB A 11 A = 20º, b =5, c = Use Law of Cosines to solve for one of the other angles. 4. Find the last angle. 20º 144.8º B = 180 – – 20 B = 15.2º “SSS: Second step sines”  SAS second step cosines”  SAS second step cosines” 15.2º

SAS Case C B 6.5 5A11 a = 11, b = 5, C = 20º If you use the Law of Sines to solve for one of the other angles. Find the last angle. 20º 35.4º B = 180 – 35.4 – 20 B = 124.6º Does it make sense that the shortest side is opposite the largest angle? opposite the largest angle? “SSS: Second step sines”  SAS second step cosines”  SAS second step cosines” 124.6º

Solving a Triangle (SAS) Your turn: 5. c = ? 6. A = ? (SAS (second step NOT law a sines) 7. B = ?

Summary SSS 1. Always solve for the largest angle first 2. Use Law of Sines to find the second angle 3. Calculate third angle SAS 1. Use Law of cosines to find missing side. 2. Use Law of cosines to find the second angle 3. Calculate third angle

HOMEWORK Section 13-6 (page 892) (evens) 8-14 (3 pts each), 22, 24 (14 total points)