TBI

504

TEI

To find, the angle A B D.

As A b, 562. 2.749736

ia to a d, 533.405 . 2.727057

so is the sine of 90°. 10.000000

12.727057

to the sine A b d, 71° 39' 9.977321

then adding cbd = 56° 25'

gives the whole angle abc = 128° 4'then 90 — 71° 39'= 18° 21' = angle Aand 90 — 56° 25' = 33° 35' = angle C.

From three parts of a triangle, including at least one sideas a part given, the other three remaining parts have beenfound upon general principles; but when the triangle is right-angled, other solutions, more simple for general practice, maybe given ; these are founded upon the following proportions:

Let abc be a right-angled triangle, right-angled at b ;upon A b take any radius, A d ; with a as a centre, and theradius A D, describe an arc, D G, cutting A C at G ; draw D eand G F parallel to b c, cutting a C at e, and A b at F ; then,by the definitions given, a C or a d is the radius, e d is thetangent, g f the sine, a f the cosine, and a e the secant ofthe arc C d. Now the sine, cosine, tangent, &c. of an arc,being the same as the sine, cosine, tangent, &c. of an angledrawn from the centre through the extremities of the arc,the following proportions result by similar triangles, A d eand abc.

ad :de ::ab :bc; that is, as a radiusis to the tangent of angle, A,so is the base, a b,to the perpendicular, b c.

By similar triangles, A F G and ABC,af:fg ::ab:bc; that is,

As the cosine of angle, A,is to the sine,so is the base, A B,to the perpendicular, b c.

So that having the angle, A, and the side, A b, given, theside b C may be found by either of these analogies ; or,having the sides or legs a b and b c, angle A can easilybe found.

By similar triangles, A e p and A c B,ae :ed ::ac :cb; that is,

As the secant (

is to the tangent

so is the hypothenuse, A c,

to the perpendicular, c b ;

or, by similar triangles, agf and Ac B,

ag:gf::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, theperpendicular, b c, may be found; or, the angle, a, may befound from having the perpendicular and the hypothenusegiven.

We shall now proceed to show the use of these analogiesby examples.

In making any side radius, the other sides are sines, ortangents, of their opposite angles; when either of the legs,or sides, containing the right angle, is made radius, the otherleg becomes a tangent of the opposite angle, and the hypo- |thenuse the secant of that angle, and not of an oppositeangle. ]

When the hypothenuse is made radius, the three s ' e u30sines of their opposite angles ; in this case the hypo 1of the sine 90° is eqiral to the radius, and is therefore astant quantity, its logarithm being 10.00000. on

The radius, sine, tangent, or secant, being wr ‘ tten m g ofany side, or supposed to be written, is called by the n athat side. Then,

As the name of any sideis to the name of any other side,so is the formerto the latter side.

And the same analogy obtains on the contrary. . _] e ,

In analogies for finding the parts of a right-angledone of the terms, or names, may always be the radius,will lessen the labour of the operation.

To exemplify what has been said: , \ e g,

■ If radius, or r, be written upon the leg a b, the ot #0 tb c, will be the tangent of angle A, and a c will be t ie vn jf;e9of angle A. Therefore upon bc write t’A, which ^ c ;the tangent of A, which will be the name of the si .^jggsalso upon the hypothenuse a c, write sec’ a, which s 'othe secant of angle A, or name of the hypothenuse A C- { j ieAgain, when the leg b c is made radius, a b beco t j,g

tangent of the opposite angle, C, and the hypothen . g

secant of angle C. Therefore, upon b C write r, and u P. g thewrite t’ C, and upon A C write sec’ C; then ra ^ US -^ e a ftname of the side b c, tangent of c is the name of the si ^ ^and secant of Q is the name of the hypothenuse, or si < t j, eLastly, when the hypothenuse A c is made ra l ^ (r \ e a ilegs a b and b c become the sines of their opp osll f . ° s inttherefore, if r be written upon A c, write s’ C, sign'v^^ i,of angle c. upon A b, and s’ a, signifying sine of naB1 eupon A b ; then radius is the name of A c, sine of ct 1of A b, and sine of A the name of bc. . gn pai'tft

It must be remembered, that whatever be the g' v aa <rledand whatever the parts required, any side of the ng n g a r6triangle may be made radius, except when the two 0 dgiven to find the acute angles, and this will furnish a _^ g geof proving the result. Therefore, if two of tbe b S 0 on e oigiven, radius, being always constant, must always to fin<5the parts concerned ; whence, if the two legs be gi vthe angles, one of the legs must be made radius.

To find a Side.

As the term, or name, on the given sideis to that on the required side,so is the given sideto the side required.

And to find an Angle .

As the side made radiusis to the other given side,so is radius

■*i»

tli 0

or

to the term or name upon that side.

Note. — From this property of a plane trianthree angles are together equal to two right ang le 1the following very useful corollaries arise. _ e gi' ,e °!

Corollary I.—When two angles of * triangl ethe third is also said to be given ; for it is the 9 g uof the other two, and may be found by subtractionfrom 180°. .

Corollary 2 .—When one angle of a triangle t jj e gi v " esum of the other two may bb found, by subtractionangle from two right angles, or 180°. , j-igh*’ i e ;

Corollary 3.—If one angle of a triangleother two are acute, and together make anotl> er ^gt ' s j ia tand, if one of the acute angles be given, the ^ 0 r "given, being the complement of the other giveoit wants of 90°.