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

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.

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