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Law of Sines: Triangle Area and Solving

Learn how to find the area of a triangle using the Law of Sines and solve triangles with side-angle-side (AAS) and angle-side-angle (ASA) information. Examples provided.

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Law of Sines: Triangle Area and Solving

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  1. 13-5 The Law of Sines Warm Up Lesson Presentation Lesson Quiz Holt McDougal Algebra 2 Holt Algebra 2

  2. Warm Up Find the area of each triangle with the given base and height. 1. b =10, h = 7 2.b = 8, h = 4.6 35 units2 18.4 units2 Solve each proportion. 3. 4. 10 28.5 5. In ∆ABC, mA = 122° and mB =17°. What is the mC ? 41°

  3. Objectives Determine the area of a triangle given side-angle-side information. Use the Law of Sines to find the side lengths and angle measures of a triangle.

  4. A sailmaker is designing a sail that will have the dimensions shown in the diagram. Based on these dimensions, the sailmaker can determine the amount of fabric needed. The area of the triangle representing the sail is Although you do not know the value of h, you can calculate it by using the fact that sin A = , or h = c sin A.

  5. Area = Area = Write the area formula. Substitute c sin A for h. This formula allows you to determine the area of a triangle if you know the lengths of two of its sides and the measure of the angle between them.

  6. Helpful Hint An angle and the side opposite that angle are labeled with the same letter. Capital letters are used for angles, and lowercase letters are used for sides.

  7. Area = ab sin C Example 1: Determining the Area of a Triangle Find the area of the triangle. Round to the nearest tenth. Write the area formula. Substitute 3 for a, 5 for b, and 40° for C. Use a calculator to evaluate the expression. ≈ 4.820907073 The area of the triangle is about 4.8 m2.

  8. Area = ac sin B Check It Out! Example 1 Find the area of the triangle. Round to the nearest tenth. Write the area formula. Substitute 8 for a, 12 for c, and 86° for B. Use a calculator to evaluate the expression. ≈ 47.88307441 The area of the triangle is about 47.9 m2.

  9. bc sin A = ac sin B = ab sin C bcsin A ac sin B ab sin C = = abc abc abc sin A = sin B = sin C b c a The area of ∆ABC is equal to bc sin A or ac sin B or ab sin C. By setting these expressions equal to each other, you can derive the Law of Sines. Multiply each expression by 2. bc sin A = ac sin B = ab sin C Divide each expression by abc. Divide out common factors.

  10. The Law of Sines allows you to solve a triangle as long as you know either of the following: 1. Two angle measures and any side length–angle-angle-side (AAS) or angle-side-angle (ASA) information 2. Two side lengths and the measure of an angle that is not between them–side-side-angle (SSA) information

  11. Example 2A: Using the Law of Sines for AAS and ASA Solve the triangle. Round to the nearest tenth. Step 1. Find the third angle measure. mD + mE + mF = 180° Triangle Sum Theorem. Substitute 33° for mD and 28° for mF. 33° + mE + 28° = 180° mE = 119° Solve for mE.

  12. sin F sin D sin F sin E = = d e f f sin 28° sin 28° sin 119° sin 33° = = e d 15 15 15 sin 33° 15 sin 119° d = e = sin 28° sin 28° d ≈ 17.4 e ≈ 27.9 Example 2A Continued Step 2 Find the unknown side lengths. Law of Sines. Substitute. Cross multiply. e sin 28° = 15 sin 119° d sin 28° = 15 sin 33° Solve for the unknown side.

  13. r Q Example 2B: Using the Law of Sines for AAS and ASA Solve the triangle. Round to the nearest tenth. Step 1 Find the third angle measure. Triangle Sum Theorem mP = 180° – 36° – 39° = 105°

  14. 10 sin 36° 10 sin 39° q= r= ≈ 6.1 ≈ 6.5 sin 105° sin 105° r Q sin Q sin R sin P sin P = = p q p r sin 39° sin 36° sin 105° sin 105° = = r q 10 10 Example 2B: Using the Law of Sines for AAS and ASA Solve the triangle. Round to the nearest tenth. Step 2 Find the unknown side lengths. Law of Sines. Substitute.

  15. Check It Out! Example 2a Solve the triangle. Round to the nearest tenth. Step 1 Find the third angle measure. mH + mJ + mK = 180° Substitute 42° for mH and 107° for mJ. 42° + 107° + mK = 180° mK = 31° Solve for mK.

  16. sin J sin H sin H sin K = = h k h j sin 42° sin 107° sin 31° sin 42° = = k h 8.4 12 12 sin 42° 8.4 sin 31° h = k = sin 107° sin 42° h ≈ 8.4 k ≈ 6.5 Check It Out! Example 2a Continued Step 2 Find the unknown side lengths. Law of Sines. Substitute. Cross multiply. 8.4 sin 31° = k sin 42° h sin 107° = 12 sin 42° Solve for the unknown side.

  17. Check It Out! Example 2b Solve the triangle. Round to the nearest tenth. Step 1 Find the third angle measure. Triangle Sum Theorem mN = 180° – 56° – 106° = 18°

  18. sin M sin P sin N sin M = = sin 56° sin 106° n m m p = p 4.7 sin 106° sin 18° 1.5 sin 106° 4.7 sin 56° = m= p= ≈ 4.7 ≈ 4.0 m 1.5 sin 18° sin 106° Check It Out! Example 2b Solve the triangle. Round to the nearest tenth. Step 2 Find the unknown side lengths. Law of Sines. Substitute.

  19. When you use the Law of Sines to solve a triangle for which you know side-side-angle (SSA) information, zero, one, or two triangles may be possible. For this reason, SSA is called the ambiguous case.

  20. Solving a Triangle Given a, b, and mA

  21. Remember! When one angle in a triangle is obtuse, the measures of the other two angles must be acute.

  22. C b a A B c Example 3: Art Application Determine the number of triangular banners that can be formed using the measurements a = 50, b = 20, and mA = 28°. Then solve the triangles. Round to the nearest tenth. Step 1 Determine the number of possible triangles. In this case, A is acute. Because b < a; only one triangle is possible.

  23. Example 3 Continued Step 2 Determine mB. Law of Sines Substitute. Solve for sin B.

  24. m B = Sin-1 Example 3 Continued Let B represent the acute angle with a sine of 0.188. Use the inverse sine function on your calculator to determine mB. Step 3 Find the other unknown measures of the triangle. Solve for mC. 28° + 10.8° + mC = 180° mC = 141.2°

  25. Example 3 Continued Solve for c. Law of Sines Substitute. Solve for c. c ≈ 66.8

  26. Check It Out! Example 3 Determine the number of triangles Maggie can form using the measurements a = 10 cm, b = 6 cm, and mA =105°. Then solve the triangles. Round to the nearest tenth. Step 1 Determine the number of possible triangles. In this case, A is obtuse. Because b < a; only one triangle is possible.

  27. Check It Out! Example 3 Continued Step 2 Determine mB. Law of Sines Substitute. Solve for sin B. sin B ≈ 0.58

  28. m B = Sin-1 Check It Out! Example 3 Continued Let B represent the acute angle with a sine of 0.58. Use the inverse sine function on your calculator to determine m B. Step 3 Find the other unknown measures of the triangle. Solve for mC. 105° + 35.4° + mC = 180° mC = 39.6°

  29. Check It Out! Example 3 Continued Solve for c. Law of Sines Substitute. Solve for c. c ≈ 6.6 cm

  30. Lesson Quiz: Part I 1. Find the area of the triangle. Round to the nearest tenth. 17.8 ft2 2. Solve the triangle. Round to the nearest tenth. a 32.2; b  22.0; mC = 133.8°

  31. Lesson Quiz: Part II 3. Determine the number of triangular quilt pieces that can be formed by using the measurements a = 14 cm, b = 20 cm, and mA = 39°. Solve each triangle. Round to the nearest tenth. 2; c1 21.7 cm; mB1≈ 64.0°; mC1≈ 77.0°; c2≈ 9.4 cm; mB2≈ 116.0°; mC2≈ 25.0°

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