Accuracy of Results Using Approximations in .NET

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3.5 Accuracy of Results Using Approximations
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When using approximate numbers, the results need to be rounded. In this chapter, we will report angles to the nearest degree and lengths to the nearest unit. If a problem has intermediate values to be computed, wait to round numbers until the final result is found. Each intermediate value should have at least one more digit than the final result is to have so that each rounding does not directly involve the unit of accuracy.
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CHAPTER 3 Trigonometric Functions of an Acute Angle
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3.6 Selecting the Function in Problem Solving
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In finding a side of a right triangle when an angle and a side are known, there are two trigonometric functions which can be used, a function and its reciprocal. When manually solving the problem, the choice is usually made so the unknown side is in the numerator of the fraction. This is done so that the operation needed to solve the equation will be multiplication rather than division. Most tables of values of trigonometric functions do not include values for secant and cosecant. You will need to use cosine instead of secant and sine instead of cosecant, when your tables only include values for sine, cosine, tangent, and cotangent. When a calculator is used, the function selected is sine, cosine, or tangent, since these functions are represented by keys on the calculator.
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EXAMPLE 3.4 A support wire is anchored 12 m up from the base of a flagpole, and the wire makes a 15 angle with the ground. How long is the wire From Fig. 3.4, it can be seen that both sin 15 and csc 15 involve the known length 12 m and the requested length x. Either function can be used to solve the problem. The manual solution, that is, using tables and not a calculator, is easier using csc 15 , but not all trigonometric tables list values for secant and cosecant. The calculator solution will use sin 15 since there is no function key for cosecant.
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Fig. 3.4
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Manual Solution csc 15 x x x x x 12 12 csc 15 12(3.86) 46.32 46 m or sin 15 x x x x 12 x 12 sin 15 12 0.26 46.15 46 m
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Calculator Solution sin 15 x x x x 12 x 12 sin 15 12 0.258819 46.3644 46 m
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The wire is 46 m long.
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In each solution, the result to the nearest meter is the same, but the results of the computations are different because of the rounding used in determining the value of the function used. Rounding to a few decimal places, as in the table provided in this section, often leads to different computational results. Using the fourdecimal-place tables in App. 2 will result in very few situations where the choice of functions affects the results of the computation. Also, when these tables are used, the results will more frequently agree with those found using a calculator. For the problems in this chapter, a manual solution and a calculator solution will be shown and an answer for each procedure will be indicated. In later chapters, an answer for each method will be indicated only when the two procedures produce different results.
CHAPTER 3 Trigonometric Functions of an Acute Angle
The decision to use or not to use a calculator is a personal one for you to make. If you will not be able to use a calculator when you apply the procedures studied, then do not practice them using a calculator. Occasionally there will be procedures discussed that are used only with tables, and others that apply to calculator solutions only. These will be clearly indicated and can be omitted if you are not using that solution method.
3.7 Angles of Depression and Elevation
An angle of depression is the angle from the horizontal down to the line of sight from the observer to an object below. The angle of elevation is the angle from the horizontal up to the line of sight from the observer to an object above. In Fig. 3.5, the angle of depression from point A to point B is and the angle of elevation from point B to point A is . Since both angles are measured from horizontal lines, which are parallel, the line of sight AB is a transversal, and since alternate interior angles for parallel lines are equal, (See App. 1, Geometry.)
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