473. - Right-Angled Triangles: The Sides. - In right-angled triangles, when two sides are given, the third side may be found by the relation of equality which exists of the squares of the sides (Arts. 353 and 416). For example, if the sides a and b (Fig. 313) are given, c, the third side, may be computed from equation (115.) -

Section XIV Trigonometry 461

Fig. 313.

c2 = b2 + a2 . Extracting the square root, we have -

Section XIV Trigonometry 462

When the hypothenuse and one side are given, by transposition of the factors in (115.), we have -

a2 = c2 - b2;

Section XIV Trigonometry 463

(A.)

or -

b2=c2-a2;

Section XIV Trigonometry 464

(B.)

Owing to the factors being involved to the second power in this expression, the labor of computation is greater than that in a more simple method, which will now be shown.

In equation (A.) or (B.) the factors under the radical may be simplified. By equation (114.) we have -

c2-b2 = (c + b)(c-b). Therefore, equation (A.) becomes -

Section XIV Trigonometry 465

a form easy of solution.

For example: let c equal 29.732 and b equal 13.216, then we have -

29.732

13.216

The sum

= 42.948

The difference

= 16.516

By the use of a table of logarithms (Art.427) the problem may be easily solved; thus -

Log. 42.948

== 1.6329429

16.516

=1.2179049

To get the square root -

(2)2.8508478

a = 26.6332

= 1.4254239

This method is applicable to the sides of a triangle, only; for the hypothenuse it will not serve. The length of the hypothenuse as well as that of either side may, however, be obtained by proportion; provided a triangle of known dimensions and with like angles be also given.

For example: in Fig-. 314, in which the two sides a and b are known, let it be required to find c, the hypothenuse.

Draw the line D E parallel with A C, then the two triangles B D E and B AC are homologous; consequently their corresponding sides are in proportion (Art. 361). Hence, if d equals unity, we have -

d: f:: a: c, = af,

from which, when a and f are known, c is obtained by simple multiplication.

474. - Right-Angled Triangles: Trigonometrical Tables. - To render the simple method last named available, the lengths of d, e and f (Fig. 314) have been computed for triangles of all possible angles, and the results arranged in tables, termed Trigonometrical Tables. The lines d, e, and f, are known as sines, cosines, tangents, cotangents, etc., as shown in Fig. 315 - where A B is the radius of the circle B C H. Draw a line A F, from A, through any point, C, of the arc B G. From C draw CD perpendicular to A B; from B draw BE perpendicular to A B; and from G draw GF perpendicular to A G.

Section XIV Trigonometry 466

Fig. 314.

Then, for the angle FA B, when the radius A C equals unity, CD is the sine; AD the cosine; DB the versed sine; BE the tangent; GF the cotangent; AE the secant; and A Fthe cosecant.

But if the angle be larger than one right angle, yet less than two right angles, as BAH, extend HA to K and E B to K, and from H draw H J perpendicular to A J.

Trigonometrical Tables

Then, for the angle BAH, when the radius A H equals unity, H J is the sine; A J the cosine; BJ the versed sine; B K the tangent; and A K the secant.

When the number of degrees'contained in a given angle is known, the value of the sine, cosine, etc., corresponding to that angle, may be found in a table of Natural Sines, Cosines, etc. Or, the logarithms of the sines, cosines, etc., may be found in logarithmic tables.

Trigonometrical Tables 467

Fig. 315.

In the absence of such a table, arid when the degrees contained in the given angle are unknown, the values of the sine, cosine, etc., may be found by computation, as follows: - Let ABC (Fig. 316) be the given angle. At any distance from B draw b perpendicular to B C. By any scale of equal parts obtain the length of each of the three lines a, b, c. Then for the angle at B we have, by proportion -

c:

b

::

1.0:

sin.

B

=b/c.

c:

a

::

1.0:

cos.

B

=a/c.

a:

b

::

1.0:

tan.

B

=b/a.

b:

a

::

1.0:

cot.

B

=a/b.

a:

c

::

1.0:

sec.

B

=c/a.

b:

c

::

1.0:

cosec

.B

=c/b.

Or, in any right-angled triangle, for the angle contained between the base and hypothenuse -

When perp. divided by hyp., the quotient equals the sine.

,,

base

,,

,,

hyp.,

,,

,,

,,

cosine.

,,

perp.

,,

,,

base,

,,

,,

,,

tangent.

,,

base

,,

,,

perp.,

,,

,,

,,

cotangent.

,,

hyp.

,,

,,

base,

,,

,,

,,

secant.

,,

hyp.

,,

,,

perp.,

,,

,,

,,

cosecant.

To designate the angle to which a trigonometrical term applies, the letter at the intended angle is annexed to the name of the trigonometrical term; thus, in the above example, for the sine of ABC we write sin. B; for the cosine, cos. B, etc.

Trigonometrical Tables 468

Fig. 316.

Equations To Right-Angled Triangles

By these proportions the two acute angles of a right-angled triangle may be computed, provided two of the sides are known. For when the perpendicular and hypoth-enuse are known, the sine and cosecant may be obtained. When the base and hypothenuse are known, the cosine and secant may be computed. And when the base and perpendicular are known, the tangent and cotangent may be computed.

Either one of these, thus obtained, shows by the trigonometrical tables the number of degrees in the angle; and, deducting the angle thus found from 900, the remainder will be the angle of the other acute angle of the triangle. For example: in a right-angled triangle, of which the base is 8 feet and the perpendicular 6 feet, how many degrees are contained in each of the acute angles?

Equations To Right Angled Triangles 469