SECTION 8.3 The Law of Cosines 573 Historical Feature The Law of Sines was known vaguely long before it was explicitly stated by Nasir Eddin (about AD 1250). Ptolemy (about AD 150) was aware of it in a form using a chord function instead of the sine function. But it was first presented in the form we use today by Regiomontanus, who wrote in 1464. The Law of Cosines appears first in Euclid’s Elements (Book II), but in a well-disguised form in which squares built on the sides of triangles are added and a rectangle representing the cosine term is subtracted. It was thus known to all mathematicians because of their familiarity with Euclid’s work. An early modern form of the Law of Cosines, that for finding the angle when the sides are known, was stated by François Viète (in 1593). The Law of Tangents (see Problem 65 in Section 8.2) has become obsolete. In the past it was used in place of the Law of Cosines, because the Law of Cosines was very inconvenient for calculation with logarithms or slide rules. Mixing of addition and multiplication is now very easy on a calculator, however, and the Law of Tangents has been shelved along with the slide rule. ‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. 8.3 Assess Your Understanding 1. Write the formula for the distance d from P x y , 1 1 1 ( ) = to P x y , . 2 2 2 ( ) = (pp. 13–15) 2. If θ is an acute angle, solve the equation cos 2 2 . θ = (pp. 493–496) (c) The total length of the trip is now 60 96 156 miles. + = The extra 6 miles will only require about 0.4 hour, or 24 minutes, more if the speed of 15 miles per hour is maintained. Now Work PROBLEM 47 Skill Building In Problems 9–16, solve each triangle. 9. 458 2 4 b A C 10. 308 4 3 a B C 11. 958 2 3 c B A 12. 208 2 5 b C A 13. 6 5 8 C A B 14. 8 5 4 C B A 15. 9 6 4 C A B 16. 3 4 4 C A B Concepts and Vocabulary 3. If three sides of a triangle are known, the Law of is used to solve the triangle. 4. Multiple Choice If one side and two angles of a triangle are known, which law can be used to solve the triangle? (a) Law of Sines (b) Law of Cosines (c) Either a or b (d) The triangle cannot be solved. 5. Multiple Choice If two sides and the included angle of a triangle are known, which law can be used to solve the triangle? (a) Law of Sines (b) Law of Cosines (c) Either a or b (d) The triangle cannot be solved. 6. True or False Given only the three sides of a triangle, there is insufficient information to solve the triangle. 7. True or False The Law of Cosines states that the square of one side of a triangle equals the sum of the squares of the other two sides, minus twice their product. 8. True or False A special case of the Law of Cosines is the Pythagorean Theorem. 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure
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