Once again we use the sum formula for cosine to get

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: SA case. 11.2 The Law of Sines 767 the Law of Sines gives sin(α) = sin(γ ) so that sin(γ ) = c sin(α) = a = 1. Here, γ = 90◦ as required. a c a a Moving along, now suppose h < a < c. As before, the Law of Sines9 gives sin(γ ) = c sin(α) . Since a h < a, c sin(α) < a or c sin(α) < 1 which means there are two solutions to sin(γ ) = c sin(α) : an a a acute angle which we’ll call γ0 , and its supplement, 180◦ − γ0 . We need to argue that each of these angles ‘fit’ into a triangle with α. Since (α, a) and (γ0 , c) are angle-side opposite pairs, the assumption c > a in this case gives us γ0 > α. Since γ0 is acute, we must have that α is acute as well. This means one triangle can contain both α and γ0 , giving us one of the triangles promised in the theorem. If we manipulate the inequality γ0 > α a bit, we have 180◦ − γ0 < 180◦ − α which gives (180◦ − γ0 ) + α < 180◦ . This proves a triangle can contain both of the angles α and (180◦ − γ0 ), giving us the sec...
View Full Document

This note was uploaded on 05/03/2013 for the course MATH Algebra taught by Professor Wong during the Fall '13 term at Chicago Academy High School.

Ask a homework question - tutors are online