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Use trig. to find the area of triangles.

Objectives. Use trig. to find the area of triangles. Use the Law of Sines to find the side lengths and angle measures of a triangle. Notes #1-3. Find the area of the triangle. Round to the nearest tenth. 2. Solve the triangle.

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Use trig. to find the area of triangles.

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  1. Objectives Use trig. to find the area of triangles. Use the Law of Sines to find the side lengths and angle measures of a triangle.

  2. Notes #1-3 • Find the area of the triangle. Round to the nearest tenth. 2. Solve the triangle. 3. Triangular banners can be formed using the measurements a = 48, b = 28, and mA = 35°. Solve the triangle (nearest tenth).

  3. 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 (round). ≈ 4.82

  4. 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.

  5. 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. Substitute 33° for mD and 28° for mF. 33° + mE + 28° = 180° mE = 119° Solve for mE.

  6. 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.

  7. 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°

  8. 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.

  9. m B = Sin-1 Example 3: Art Application Triangular banners can be formed using the measurements a = 50, b = 20, and mA = 28°. Solve the triangle (nearest tenth). Step 1 Determine mB.

  10. Example 3 Continued Step 3 Find the other unknowns in the triangle. 28° + 10.8° + mC = 180° mC = 141.2° Solve for c. Solve for c. c ≈ 66.8

  11. Notes #1-3 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°

  12. Example 3: Art Application Triangular banners can be formed using the measurements a = 48, b = 28, and mA = 35°. Solve the triangle (nearest tenth).

  13. Notes #3 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°

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

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