## Tape Sine Method

The tape-sine method uses a combination of distances measured by a tape and the sine trigonometric function. This method is not limited to 90° and can also be used to lay out an angle or measure the angle between two existing lines. A review of the three commonly used trigonometric functions will make this method clear.

Trigonometric functions are based on the principle that one unique ratio exists between the lengths of any two sides for angles B and C, Figure 13.4. Because a right triangle has three sides, each angle (B and C) has six possible combinations of two sides. Each of these combinations has been given a name. The three common ratios are:

Length of opposite side (b) Length of hypotenuse (a) Length adjacent side (c) Length of hypotenuse (a) Length opposite side (b) Length adjacent side (c)

The same relationships are true for angle C. Each of these functions forms an equation with three variables—the function of the angle and the lengths of two sides. If any two of the variables are known, the third can be determined.

In the tape-sine method, only the sine function is used. The procedure for laying out an angle (0) is slightly different from the procedure for measuring an existing angle.

The procedure for measuring an existing angle will be explained first. Using the example illustrated in Figure 13.5, the task is to determine the angle formed by BAC. The first step is to mark an equal distance along each side AB and AC. The FIGURE 13.5. Example of tape-sine method.

next step is to measure the distance BC. The last step is to form two right triangles by drawing a line from corner A to the mid point of line BC.

The distances BC/2 and AC are two sides of a right triangle. For this example we will assume that the distance BC = 61.8 ft. The angle for either triangle can be found by using the Sine equation.

Length of opposite side (b)

Length of hypotenuse (a) Distance BC/2 _ 30.9 ft Distance AC = 100 ft

The value 0.309 is the sine ratio of angle A. The next step is to determine the angle having a sine ratio of 0.309.

To determine the angle using a calculator, enter 0.309 and the inverse of the sine function. The angle can also be determined by consulting Table 13.1.

 Angle 0° Sine of the angle Angle 0° Sine of the angle Angle 0° Sine of the angle 0 0.000 31 0.515 61 0.875 1 0.017 32 0.530 62 0.883 2 0.035 33 0.545 63 0.891 3 0.052 34 0.559 64 0.899 4 0.070 35 0.574 65 0.906 5 0.087 36 0.588 66 0.914 6 0.105 37 0.602 67 0.921 7 0.122 38 0.616 68 0.927 8 0.139 39 0.629 69 0.934 9 0.156 40 0.643 70 0.940 10 0.174 41 0.656 71 0.946 11 0.191 42 0.669 72 0.951 12 0.208 43 0.682 73 0.956 13 0.225 44 0.695 74 0.961 14 0.242 45 0.707 75 0.966 15 0.259 46 0.719 76 0.971 16 0.276 47 0.731 77 0.974 17 0.292 48 0.743 78 0.978 18 0.309 49 0.755 79 0.982 19 0.326 50 0.766 80 0.985 20 0.342 51 0.777 81 0.988 21 0.358 52 0.788 82 0.990 22 0.375 53 0.799 83 0.993 23 0.391 54 0.809 84 0.995 24 0.407 55 0.819 85 0.996 25 0.423 56 0.829 86 0.998 26 0.438 57 0.839 87 0.999 27 0.454 58 0.848 88 0.999 28 0.469 59 0.857 89 1.000 29 0.485 60 0.866 90 1.000 30 0.500

FIGURE 13.6. Laying out an angle by the tape-sine method. Either source should give an angle of 18°. Because we used the distance BC/2 to solve for the angle, the angle CAB is actually 18° x 2, or 36°.

The same procedures can be used to lay out an angle. For example, suppose you need to establish a fence at 40° to an existing fence. How would the tape-sine method be used? Study Figure 13.6.

The principles involved are basic trigonometry. It takes two points to establish a line and to locate a point either three dimensions or an angle and one dimension must be known. To establish the line AC so that it forms a 40° angle with the base line AB, three dimensions are used to locate point C. Two of the dimensions are the distance selected by the person laying out the angle. In this example 50 ft is used. The third dimension is the distance from point B to point C. To determine this dimension mentally split the triangle CAB into two right triangles (BAD and CAD), solve for the length of the opposite side of either, and then double this distance. The doubled distance is distance from B to C. The distance A to C is the same as A to B. Using point B and the length BS scribe a short arc in the vicinity of point C. Using point A and the length AB scribe another arc. The intersection of the two arcs is point C. A line from point A through C will establish the fence at the correct angle.

To determine the distance BC/2 the sine trig functions is rearranged to solve for the length of the opposite side:

Opposite

Hypotenuse Opposite = Sine A x Hypotenuse = Sine 20° x 50.0 ft = 0.3420... x 50.0 ft = 17.1010... ft 17.1010... x 2 = 34.2020... or 34.2 ft

To lay out the angle, mark an arc, with a radius of 34.2 ft, from point B and another arc, with a radius of 50, from point A, and set a pin at the intersection of

FIGURE 13.7. Irregular-shaped field.

the two arcs (point C). A line drawn from point A through point C will establish a line at a 40° angle to the base line (AB). In this method, the marking of two arcs could also be eliminated by using two tape measures.