AO:OF::AC:CB; that is, As radius is to the sine of angle, A, so is the hypothenuse, A C, to the perpendicular, C B.
So that having the angle, A, and the hypothenuse given, the perpendicular, B c, may be found ; or, the angle, A, may be found from having the perpendicular and the hypothenuse given.
We shall now proceed to show the use of these analogies by examples.
In making any side radius, the other sides are sines, or tangents, of their opposite angles ; when either of the legs, or sides, containing the right angle, is made radius, the other leg becomes a tangent of the opposite angle, and the hypu thenuse the secant of that angle, and not of an opposite angle.
When the hypothenuse is made radius, the three sides are sines of their opposite angles ; in this case the hypothenuse of the sine is equal to the radius, and is therefore a con stant quantity, its logarithm being 10.00000.
The radius, sine, tangeht, or secant, being written upon any side, or supposed to be written, is called by the name of that side. Then, As the name of,any side is to the name of any other side, so is the former to the latter side.
And the same analogy obtains on the contrary.
In analogies for finding the parts of a right-angled triangle, one of the terms, or names, may always be the radius, which will lessen the labour of the operation.
To exemplify what has been said : If radius, or r, be written upon the leg A a, the other leg, B c, will be the tangent of angle A, and A c will be the secant of angle A. Therefore upon B c write t'A, which signifies the tangent of A, which will be the name of the side B C; also upon the hypothenuse A c, write see' A, which signifies the secant of angle A, or name of the hypothenuse A c.
Again, when the leg B C is made radius, A n becomes the tangent of the opposite angle, c, and the hypothenuse the secant of angle c. Therefore, upon 13 c write r, and upon A B write t' c, and upon A c write see' c ; then radius is the name of the side B C, tangent of c is the name of the side A B, and secant of c is the name of the hypothenuse, or side, A C.
Lastly, when the hypothenuse A C is made radius, the legs A B and B C become the sines of their opposite angles; therefore, if r be written upon A C, write s' c, signifying sine of angle c, upon A B, and s' A, signifying sine of angle A, upon A B; then radius is the name of A c, sine of c the name of A B, and sine of A the name of n C.
It must be remembered, that whatever be the given parts, and whatever the parts required, any side of the right-angled triangle may be made radius, except when the two legs are given to find the acute angles, and this will furnish a method of proving the result. Therefore, if two of the sides be
given, radius, being always constant, must always be one of the parts concerned ; whence, if the two legs be given to find the angles, one of the legs must he made radius.
To find a Side.
As the term, or name on the given side is to that on the required side, so is the given side to the side required.
And to find an Angle.
As the side made radius is to the other given side, so is radius to the term or name upon that side.
Note.—From this property of a plane triangle, that the three angles are together equal to two right angles, or 180°, the following very useful corollaries arise.
Corollary I—When two angles of a triangle are given, the third is also said to be given ; for it is the supplement of the other two, and may be found by subtracting their sum from 180°.
Corollary 2.—When one angle of a triangle is given, the sum of the other two may be found, by subtracting the given angle from two right angles, or Corollary 3.—If one angle of a triangle be right, the other two are acute, and together make another right angle; and, if one of the acute angles be given, the other is also given, being the complement of the other given one, or what it wants of 90°.
Geometrically.
Draw the indefinite line n c, and from the point c, with the chord of 60°, describe an arc, upon which lay off the quantity of the angle c, 55° 30'; then place the hypothenuse 121 equal parts from c to A, and from A let fall a perpen dicular to B. Measure the sides A B and e c on the scale from which A c was taken.
Example 2.—In the triangle A B c, right-angled at a, sup pose the base, B c, 3741 yards, and the angle, A, 52° 8'; required the other side, A B, and the hypothenuse, A C.
Example 3.—Suppose a ship sail s. w. by w. until she has made 409 miles of southing ; required the distance sailed, and also how far she is west from the meridian of the place sailed from.
Example the sun's altitude to be 30° 45', and the shadow of a tree at the same time to fall 70 feet 3 inches distant from the tree, on the horizontal plane; what is the height of the tree, and what will be the length of a rope to reach from the extremity of the shadow to the top of the tree ?