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Pre-Calc Exam Notes 8

Pre-Calc Exam Notes 8 - United States and one in Germany...

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8 Chapter 1 Right Triangle Trigonometry §1.2 Example 1.5 A C B 4 3 5 For the right triangle ABC shown on the right, fnd the values o± all six trigono- metric ±unctions o± the acute angles A and B . Solution: The hypotenuse o± ABC has length 5. For angle A , the opposite side BC has length 3 and the adjacent side AC has length 4. Thus: sin A = opposite hypotenuse = 3 5 cos A = adjacent hypotenuse = 4 5 tan A = opposite adjacent = 3 4 csc A = hypotenuse opposite = 5 3 sec A = hypotenuse adjacent = 5 4 cot A = adjacent opposite = 4 3 For angle B , the opposite side AC has length 4 and the adjacent side BC has length 3. Thus: sin B = opposite hypotenuse = 4 5 cos B = adjacent hypotenuse = 3 5 tan B = opposite adjacent = 4 3 csc B = hypotenuse opposite = 5 4 sec B = hypotenuse adjacent = 5 3 cot B = adjacent opposite = 3 4 Notice in Example 1.5 that we did not speci±y the units ±or the lengths. This raises the possibility that our answers depended on a triangle o± a specifc physical size. For example, suppose that two di±±erent students are reading this textbook: one in the
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Unformatted text preview: United States and one in Germany. The American student thinks that the lengths 3, 4, and 5 in Example 1.5 are measured in inches, while the German student thinks that they are measured in centimeters. Since 1 in ≈ 2.54 cm, the students are using triangles o± di±±erent physical sizes (see Figure 1.2.1 below, not drawn to scale). A B C 3 4 5 (a) Inches A ′ B ′ C ′ 3 4 5 (b) Centimeters A A ′ B B ′ C C ′ (c) Similar triangles Figure 1.2.1 △ ABC ∼ △ A ′ B ′ C ′ I± the American triangle is △ ABC and the German triangle is △ A ′ B ′ C ′ , then we see ±rom Figure 1.2.1 that △ ABC is similar to △ A ′ B ′ C ′ , and hence the corresponding angles...
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