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7.2 The Law of Cosines and Area Formulas

7.2 The Law of Cosines and Area Formulas. SAS or SSS forms a unique triangle Triangle Side Restriction In any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. 7.2 Derivation of the Law of Cosines.

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7.2 The Law of Cosines and Area Formulas

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  1. 7.2 The Law of Cosines and Area Formulas • SAS or SSS forms a unique triangle • Triangle Side Restriction • In any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.

  2. 7.2 Derivation of the Law of Cosines Let ABC be any oblique triangle drawn with its vertices labeled as in the figure below. The coordinates of point A become (c cos B, c sin B). Figure 10 pg 10-28

  3. 7.2 Derivation of the Law of Cosines Point C has coordinates (a, 0) and AC has length b. This result is one form of the law of cosines. Placing A or C at the origin would have given the same result, but with the variables rearranged.

  4. 7.2 The Law of Cosines The Law of Cosines In any triangle ABC, with sides a, b, and c,

  5. 7.2 Using the Law of Cosines to Solve a Triangle (SAS) Example Solve triangle ABC if A = 42.3°, b = 12.9 meters, and c = 15.4 meters. Solution Start by finding a using the law of cosines.

  6. 7.2 Using the Law of Cosines to Solve a Triangle (SAS) B must be the smaller of the two remaining angles since it is opposite the shorter of the two sides b and c. Therefore, it cannot be obtuse. Caution If we had chosen to find C rather than B, we would not have known whether C equals 81.7° or its supplement, 98.3°.

  7. 7.2 Using the Law of Cosines to Solve a Triangle (SSS) Example Solve triangle ABC if a = 9.47 feet, b =15.9 feet, and c = 21.1 feet. Solution We solve for C, the largest angle,first. If cos C < 0, then C will be obtuse.

  8. 7.2 Using the Law of Cosines to Solve a Triangle (SSS) Verify with either the law of sines or the law of cosines that B  45.1°. Then,

  9. 7.2 Summary of Cases with Suggested Procedures

  10. 7.2 Summary of Cases with Suggested Procedures

  11. 7.2 Area Formulas • The law of cosines can be used to derive a formula for the area of a triangle given the lengths of three sides known as Heron’s Formula. Heron’s Formula If a triangle has sides of lengths a, b, and c, and if the semiperimeter is Then the area of the triangle is

  12. 7.2 Using Heron’s Formula to Find an Area Example The distance “as the crow flies” from Los Angeles to New York is 2451 miles, from New York to Montreal is 331 miles, and from Montreal to Los Angeles is 2427 miles. What is the area of the triangular region having these three cities as vertices? (Ignore the curvature of the earth.) Solution

  13. 7.2 Area of a Triangle Given SAS • The area of any triangle is given by A = ½bh, where b is its base and h is its height. Area of a Triangle In any triangle ABC, the area A is given by any of the following:

  14. 7.2 Finding the Area of a Triangle (SAS) Example Find the area of triangle ABC in the figure. Solution We are given B = 55°, a = 34 feet, and c = 42 feet.

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