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CHAPTER VII

ELEMENTS OF TRIGONOMETRY

TRIGONOMETRY is a very important part of the science of mathematics, and deals with the determination of angles and the solution of triangles. In order to fully understand the subjects treated of in the following, it is necessary that the reader is fully familiar with the usual methods of designating the measurements or sizes of angles. While mathematicians employ also another method, in mechanics angles are measured in degrees and subdivisions of a degree, called minutes. The minute is again subdivided into seconds, but these latter subdivisions are so small as to permit of being disregarded in general practical machine design.

A degree is 1-360 part of a circle, or, in other words, if the circumference of a circle is divided into 360 parts, then each part is called one degree.

If two lines are drawn from the center of the circle to the ends of the small circular arc which is 1-360 part of the circumference, then the angle between these two lines is a 1-degree angle. A quarter of a circle or a 90-degree angle is called a right angle. The meaning of obtuse and acute angles has already been explained in Chapter II. Any angle which is not a right angle is called an oblique angle.

A minute is 1-60 part of a degree, and a second 1-60 part of a minute. In other words, one circle = 360 degrees, one degree = 60 minutes, and one minute = 60 seconds. The sign () is used for indicating degrees; the sign (') indicates minutes, and the sign (") seconds. A common abbreviation for degree is "deg."; for minute, "min."; and for second, "sec."

Two angles are equal when the number of degrees they contain is the same. If two angles are both 30 degrees, they are equal, no matter how long the sides of the one may be in relation to the other.

Of all triangles, the right-angled triangle occurs most frequently in machine design. A right-angled triangle is one having the angle between two sides a right angle; the angles between the other sides may be of any size. In the calculations involved in solving right-angled triangles, a useful application of the squares and square roots of numbers is also presented. Assume that the lengths of the sides of a right-angled triangle, as shown in Fig. 88, are 5 inches, 4 inches, and 3 inches, respectively. Then

52 = 42 + 32, or 25 = 16 + 9.

This relationship between the three sides in a right-angled triangle holds good for all right-angled triangles. The square of the side opposite the right angle equals the sum of the squares of the sides including the right angle. Assume, for example, that the lengths of the two sides including the right angle in a right-angled triangle are 12

and 9 inches long, respectively, as shown in Fig. 89, and that the side opposite the right angle, the hypotenuse, is to be found. We then first square the two given sides, and from our rule, just given, we have that the sum of the squares equals the square of the side to be found. The square root

-FIND HYPOTENUSE--

FIG. 88.

FIG. 89.

of the sum must then equal the side itself. Carrying out this calculation we have:

122 +92 144 +81

✓ 225

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= 225

15 inches = length of hypotenuse. Similar methods may be employed for finding any of the sides in a right-angled triangle if two sides are given. If the hypotenuse were known to be 15 inches, and one of the sides including the right angle 9 inches, as shown at D in Fig. 90, then the other side including the right angle can be found. In this case, however, we must subtract the square of the known side including the right

angle from the square of the hypotenuse to obtain the square of the remaining including side. We, therefore, have:

15292 225 81 144

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✓ 144 12 inches-length of unknown side.

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In the same way, if the lengths 15 and 12 were

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known, we could find the side AC, as shown at E, Fig. 90:

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From these examples we may formulate rules and general formulas for the solution of rightangled triangles when two sides are known. In Fig. 91, at F, the square of AB plus the square of AC equals the square of BC; the square of BC minus the square of AC equals the square of AB; and the square of BC minus the square of AB

equals the square of AC. These rules written as general formulas would take the form:

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From these formulas we have, by extracting the square root on each side of the equal sign:

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These formulas make it possible to find the third side when two sides are given, no matter what the

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numerical values of the length of the sides may be. Assume AB = 12, and BC= 20; find AC. According to the formula:

AC: ✓ 202 122 ✓ 400

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144 =

✓ 256 = 16.

Assume that AB = 15 and AC 20. Find BC.

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The rules and formulas given make it possible to find the length of the sides in a right-angled triangle. To find the angles, however, use must be

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