Trigonometry
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Trigonometry Reference Home Back to Thinkquest Contents
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Definitions of Trigonometric Functions Trigonometric Functions of Sums and Differences Inverse Trigonometric Functions
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Application Problem Using the given information, and knowledge of ambiguity, solve the triangle(s) and find haw many triangles can be made.
Information: Ambiguity is when the given information provides more then one answer. When given an angle and two sides, that does not mean that only one triangle can be made using the measurements. There could be two triangles, or maybe, none at all. For this problem, we can see how many triangles can be formed and then solve each one. Solution: To start the problem, assume that two triangles can be made. Create a chart for each triangle and plug in the information. Both <As in the triangles are 30 because that is given. As you calculate more measurements, add them to the chart. Triangle One Triangle Two <A = 30° <A = 30° Use the Law of Sines to find <B for the first triangle. SinA/a = SinC/c Sin 30°/4 = SinC/5 SinC = 0.625 <C = 38.7° For triangle two, <C would be 180° - 38.7° because the Sine is also positive in the second quadrant. That would make <C = 141.3°. Now look back at <A. If <A is 30° and <C is 141.3°, is there room for the third angle? Yes! When the two angles for triangle two are added, it is less than 180° and <B can easily be calculated to be 8.7°. Therefore, two triangles can be made. Add to the chart of the two triangles all the information we have. Triangle One Triangle Two <A = 30° <A = 30° <B = 111.3° <B = 8.7° <C = 38.7° <C = 141.3° Now that we have all the angles, we can solve for all the sides. Triangle One: SinA/a = SinB/b Sin30°/4 = Sin 111.3°/b b = 7.45 Triangle Two: SinA/a = SinB/b Sin30°/4 = Sin8.7°/b b = 1.21 Answers: Two triangles can be made Triangle One: Triangle Two: <A = 30° <A = 30° <B = 111.3° <B = 8.7° <C = 38.7° <C = 141.3° a = 4 a = 4 b = 7.45 b = 1.21 c = 5 c = 5
After finding all the angles and sides of the triangles, we can draw the picture of the diagram. We could not before, because we did not know if the measurements made any triangles.
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