 Publication details: New Zealand Government Railways Department, Wellington

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# The Uses of Trigonometry

#### The Uses of Trigonometry.

Varying in accordance with their magnitude, the angles and sides of triangles have certain relationships to one another. This relationship is called trigonometry and books of tables giving this information are used. When certain sides and angles of a triangle are known, the remaining sides or angles are readily found. I have shown in figure No. 7 the various cases, with known sides or angles in full lines, from which the remaining sides and angles are obtainable. It will be seen that practically all contingencies are met.

It is advisable to make a special note that the sum of the three angles in any triangle equal two right angles or 180 degrees, and that the adjoining angles made by a straight line meeting another straight line also equal two right angles or 180 degrees. I also bring particularly under notice the relationship of the sides of a right angled triangle i.e., a triangle having one of its angles a right angle, or 90 degrees, as this will be referred to again. This relationship is that the longer side multiplied by itself equals the sum of the two other sides when multiplied by themselves. (See Fig. No. 8.) It is interesting to note the figures 3, 4 and 5 make a right angled triangle, and should you desire to set out a right angle, such as for marking out a tennis court, these figures are easily memorised.

I mentioned previously that all measurements are horizontal, and if taken on the slope are reduced to the horizontal equivalent. In figure No. 9 is shewn an inclined measurement, and the method of arriving at the horizontal distance. The angle A of slope is observed at the instrument, the angle B is known to be a right angle, and as the sum of the angles of a triangle equal two right angles, the angle at C must be the difference between a right angle and the angle at A. We thus have a triangle ABC with all angles known, and also the side AC. The tables are looked up for the side AB, and the horizontal measurement AB is given.

In figures No. 10 and 10a are shewn the methods of ascertaining the height of inaccessible points. The observed measurements are in full lines. It will be seen the inaccessible angle and sides are computed. In arriving at the height of the mountain peak in figure 10a the side BC in the triangle ABC is first found. Then in the triangle BCD the angle BCD is known, as it equals 180 degrees less angle ACB. The angle D is a right angle, therefore the angle CBD equals a right angle less angle BCD, as the sum of the angles must equal two right angles. The height of the mountain can now be computed.