569 5.4 Solutions and Applications of Right Triangles Angles of Elevation or Depression In applications of right triangles, the angle of elevation from point X to point Y (above X) is the acute angle formed by ray XY and a horizontal ray with endpoint at X. See Figure 48(a). The angle of depression from point X to point Y (below X) is the acute angle formed by ray XY and a horizontal ray with endpoint X. See Figure 48(b). Y X Angle of depression Horizontal (b) Figure 48 Angle of elevation Horizontal X Y (a) CAUTION Be careful when interpreting the angle of depression. Both the angle of elevation and the angle of depression are measured between the line of sight and a horizontal line. To solve applied trigonometry problems, follow the same procedure as solving a triangle. Drawing a sketch and labeling it correctly in Step 1 is crucial. 123 ft a 268 409 A Figure 49 EXAMPLE 3 Finding a Length Given the Angle of Elevation At a point A, 123 ft from the base of a flagpole, the angle of elevation to the top of the flagpole is 26° 40′. Find the height of the flagpole. SOLUTION Step 1 See Figure 49. The length of the side adjacent to A is known, and the length of the side opposite A must be found. We will call it a. Step 2 The tangent ratio involves the given values. Write an equation. tan A = side opposite side adjacent Tangent ratio tan 26° 40′ = a 123 A = 26° 40′; side adjacent = 123 Step 3 a = 123 tan 26° 40′ Multiply by 123 and rewrite. a ≈12310.502218882 Use a calculator. a ≈61.8 ft Three significant digits The height of the flagpole is 61.8 ft. S Now Try Exercise 47. Solving an AppliedTrigonometry Problem Step 1 Draw a sketch, and label it with the given information. Label the quantity to be found with a variable. Step 2 Use the sketch to write an equation relating the given quantities to the variable. Step 3 Solve the equation, and check that the answer makes sense.
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