578 CHAPTER 8 Applications of Trigonometric Functions the included angle C are known. See Figure 41. Then the altitude h can be found by noting that = h a C sin so = h a C sin Using this fact in formula (1) produces ( ) = = = K bh b a C ab C 1 2 1 2 sin 1 2 sin The area K of the triangle is given by the formula Figure 41 h b a C THEOREM Area of an SAS Triangle The area K of a triangle equals one-half the product of two of its sides times the sine of their included angle. THEOREM Area of an SAS Triangle = K ab C 1 2 sin (2) Dropping altitudes from the other two vertices of the triangle leads to the following corresponding formulas: = K bc A • 1 2 sin (3) = K ac B • 1 2 sin (4) It is easiest to remember these formulas by using the following wording: Finding the Area of an SAS Triangle Find the area K of the triangle for which = = a b 8, 6, and = ° C 30 . Solution EXAMPLE 1 See Figure 42. Use formula (2) to get = = ⋅ ⋅ ⋅ ° = K ab C 1 2 sin 1 2 8 6 sin30 12 square units Now Work PROBLEMS 9 AND 17 2 Find the Area of SSS Triangles If the three sides of a triangle are known, another formula, called Heron’s Formula (named after Heron of Alexandria), can be used to find the area of a triangle. THEOREM Heron’s Formula The area K of a triangle with sides a, b, and c is ( )( )( ) = − − − K s s a s b s c (5) where ( ) = + + s a b c 1 2 . Figure 42 8 308 6 c B A
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