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The slide rule

Chapter 47: THE SOLUTION OF RIGHT-ANGLED TRIANGLES.
By Charles N. Pickworth
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About This Book

The manual defines the slide rule as an instrument for performing calculations by means of logarithms and provides a concise primer on common logarithms, mantissa and characteristic. It explains the mathematical principle that adding and subtracting logarithms yields products and quotients, and extends this to powers, roots, and trigonometric and algebraic operations. Practical instruction covers the layout and use of scales, slide and cursor, magnifying cursors and special rule types, with examples of routine procedures such as multiplication, division, square and root extraction, and proportional calculations. Descriptions of specialized instruments and applications include engineering tasks like screw-cutting and gear calculations, together with advice on accuracy and practice.

THE SOLUTION OF RIGHT-ANGLED TRIANGLES.

From the foregoing explanation of the manner of determining the trigonometrical functions of angles, the methods of solving right-angled triangles will be readily perceived, and only a few examples need therefore be given.

Let a and b represent the sides and c the hypothenuse of a right-angled triangle, and a° and b° the angles opposite to the sides. Then of the possible cases we will take

(1.) Given c and a°, to find a, b, and b°.

The angle b° = 90 − a°, while a = c sin a° and b = c sin b°. To find a, therefore, the index of S is set to c on A, and the value of a read on A opposite a° on S. In the same manner the value of b is obtained.

Ex.—Given in a right-angled triangle c = 9 ft. and a° = 30°. Find a, b, and b°.

The angle b° = 90 − 30 = 60°. To find a, set R.H. index of S to 9 on A, and over 30° on S read a = 4·5 ft. on A. Also, with the slide in the same position, read b = 7·8 ft. [7·794] on A over 60° on S.

(2.) Given a and c, to determine a°, b°, and b.

In this case advantage is taken of the fact that in every triangle the sides are proportional to the sines of the opposite angles. Therefore, as in this case the hypothenuse c subtends a right angle, of which the sine = 1, the R.H. index (or 90°) on S is set to the length of c on A, when under a on A is found a° on S. Hence b° and b may be determined.

(3.) Given a and a°, to find b, c, and b°.

Here b° = (90 − a°), and the solution is similar to the foregoing.

(4.) Given a and b, to find a°, b°, and c.

To find a°, we have tan. a° = a/b, which in the above example will be 4·5
7·8 = 0·577. Therefore, placing the slide so that the indices of T coincide with those of D, we read opposite 0·577 on D the value of a° = 30°. The hypothenuse c is readily obtained from c = a/(sin a°).