SECTION 8.2 The Law of Sines 565 (b) The time t needed for the helicopter to reach the ship from Station X-ray is found by using the formula ( )( ) = r t a Rate, Time, Distance, Then = = ≈ ≈ t a r 97.82 200 0.49 hour 29 minutes It will take about 29 minutes for the helicopter to reach the ship. Now Work PROBLEM 49 Proof of the Law of Sines To prove the Law of Sines, construct an altitude of length h from one of the vertices of a triangle. Figure 33(a) shows h for a triangle with three acute angles, and Figure 33(b) shows h for a triangle with an obtuse angle. In each case, the altitude is drawn from the vertex at B. Using either figure, = C h a sin from which = h a C sin (3) From Figure 33(a), it also follows that = A h c sin from which = h c A sin (4) From Figure 33(b), it follows that ( ) °− = = A A h c sin 180 sin ↑ ( ) °− = ° − ° = A A A A sin 180 sin180 cos cos180 sin sin which again gives = h c A sin So, whether the triangle has three acute angles or has two acute angles and one obtuse angle, equations (3) and (4) hold.As a result, the expressions for h in equations (3) and (4) are equal. That is, = a C c A sin sin from which = A a C c sin sin (5) In a similar manner, constructing the altitude ′h from the vertex of angle A, as shown in Figure 34, reveals that = ′ = ′ B h c C h b sin and sin Equating the expressions for ′h gives ′ = = h c B b C sin sin from which = B b C c sin sin (6) When equations (5) and (6) are combined, the result is the Law of Sines. ■ Figure 34 c a h9 A C B b (a) h9 c a A C B b (b) Figure 33 c a h A C b (a) B c a h A 1808 2 A C B b (b)
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