8.1 The Law of Sines Congruency and Oblique Triangles Using the Law of Sines The Ambiguous Case Area of a Triangle

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1 Chapter 8 Applications of Trigonometry The Law of Sines Congruency and Oblique Triangles Using the Law of Sines The Ambiguous Case Area of a Triangle A triangle that is not a right triangle is called an triangle. The measures of the three sides and the three angles of a triangle can be found if at least one and any other measures are known. Data Required for Solving Oblique Triangles Case 1 Case 2 side and angles are known (SAA or ASA). sides and angle not included between the two sides are known ( ). This case may lead to more than one triangle. Case 3 sides and the angle between the two sides are known ( ). Case 4 sides are known ( ). If we know three angles of a triangle, we cannot find unique side lengths since AAA assures us only of similarity, not congruence. Law of Sines In any triangle ABC, with sides a, b, and c, a b c sin A sin B sinc That is, according to the law of sines, the lengths of the sides in a triangle are to the sines of the measures of the angles opposite them. An alternative form of the law of sines is Using the Law of Sines Round answers to the nearest tenths, unless directions say otherwise. EXAMPLE 1 Applying the Law of Sines (SAA) Solve triangle ABC if A = 32.0, B = 81.8, and a = 42.9 cm.

2 8-2 Chapter 8 Applications of Trigonometry EXAMPLE 2 Applying the Law of Sines (ASA) Kurt Daniels wishes to measure the distance across the Gasconade River. See the figure. He determines that C = , A = 31.10, and b = ft. Find the distance a across the river. Reflect: In Example 2, suppose we are told that A = 31.10, b = ft, and a = ft. Can the law of sines be used to find the distance between A and B? Why or why not? Applying the Law of Sines 1. For any angle of a triangle, 0 sin 1. If sin 1, then 90 and the triangle is a right triangle. 2. sin sin 180 (Supplementary angles have the same sine value.) 3. The smallest angle is opposite the shortest side, the largest angle is opposite the longest side, and the middle-valued angle is opposite the intermediate side (assuming the triangle has sides that are all of different lengths). EXAMPLE 3 Analyzing Data Involving an Obtuse Angle Without using the law of sines, explain why A = 104, a = 26.8 m, and b = 31.3 m cannot be valid for a triangle ABC. EXAMPLE 4 Solving the Ambiguous Case (No Such Triangle) Solve triangle ABC if A = 43.5, a = 22.8 ft, and b = 42.9 ft.

3 EXAMPLE 5 Solving the Ambiguous Case (One Triangle) Solve triangle ABC, given A = 43.5, a = 22.8 in., and c = 14.8 in. 8-3 EXAMPLE 6 Solving the Ambiguous Case (Two Triangles) Solve triangle ABC if A = 43.5, a = 22.8 ft, and b = 24.9 ft. Area of a Triangle (SAS) In any triangle ABC, the area is given by the following formulas. 1 sin, 2 bc A 1 sin, 2 ab C and 1 sin 2 ac B That is, the area is half the product of the of and the of the angle between them.

4 8-4 Chapter 8 Applications of Trigonometry EXAMPLE 8 Finding the Area of a Triangle (SAS) Find the area of triangle ABC in the figure. EXAMPLE 9 Finding the Area of a Triangle (ASA) Find the area of triangle ABC in the figure.

5 Section 8.2 The Law of Cosines The Law of Cosines Derivation of the Law of Cosines Using the Law of Cosines Heron s Formula for the Area of a Triangle Derivation of Heron s Formula Triangle Side Length Restriction In any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. Law of Cosines In any triangle ABC, with sides a, b, and c, the following hold a b c bc A 2 cos, b a c ac B 2 cos, c a b 2abcosC That is, according to the law of cosines, the of a side of a triangle is equal to the of the of the other two sides, minus of those two sides and the cosine of the angle between them. Reflect: If C = 90, then what theorem does the law of cosines become? Using the Law of Cosines EXAMPLE 2 Applying the Law of Cosines (SAS) Solve triangle ABC if A = 42.3, b = 12.9 m, and c = 15.4 m. Reflect: Why couldn t we use the law of sines to solve Example 2?

6 8-6 Chapter 8 Applications of Trigonometry EXAMPLE 4 Designing a Roof Truss (SSS) Find angle B to the nearest degree, for the truss shown in the figure. Oblique Triangle Case 1: One side and two angles are known. (SAA or ASA) Case 2: Two sides and one angle (not included between the two sides) are known. (SSA) Case 3: Two sides and the included angle are known. (SAS) Case 4: Three sides are known. (SSS) Suggested Procedure for Solving Step 1 Find the remaining angle using. Step 2 Find the remaining sides using the. This is the ambiguous case. There may be no triangle, one triangle, or two triangles. Step 1 Find a second angle using the. Step 2 Find the remaining angle using. Step 3 Find the remaining sides using the. If two triangles exist, repeat Steps 2 and 3. Step 1 Find the third side using the. Step 2 Find the smaller of the two remaining angles using the. Step 3 Find the remaining angle using. Step 1 Find the largest angle using the. Step 2 Find either remaining angle using the. Step 3 Find the remaining angle using.

7 Heron s Formula for the Area of a Triangle Section 8.2 The Law of Cosines 8-7 Heron s Area Formula (SSS) If a triangle has sides of lengths a, b, and c, with semiperimeter then the area 1 s a b c, 2 of the triangle is given by the following formula. s s a s b s c EXAMPLE 5 Using Heron s Formula to Find an Area (SSS) The distance as the crow flies from Los Angeles to New York is 2451 mi, from New York to Montreal is 331 mi, and from Montreal to Los Angeles is 2427 mi. What is the area of the triangular region having these three cities as vertices? Round to the nearest hundred. (Ignore the curvature of Earth.)