Problem I.— Given the angles and hypothenuse of a right-an gled plane triangle , to find the base and perpendicular.
Example 1.—In the triangle A B C, right-angled at b, sup-P°se the angle, c, 55° 30', and the hypothenuse, a c, 121 yards;Squired the sides A b and b c.
Geometrically.
Draw the indefinite line b c, and from the point C, withhe chord of 60°, describe an arc, upon which lay off theQuantity of the angle c 55° 30'; then place the hypothenuse,~1 equal parts from c to a, and from a let fall a perpen-■cular to b. Measure the sides A b and B c on the scaler °m which A C was taken.
By Calculation.
The hypothenuse a c being radius, then A b is the sine ofangle c, and b c the sine of angle A, or cosine of angle c.^ence,
To find a b.
A 8 radius. 10.000000
18 to sine ofc, 55° 30'. 9.915994
0 18 ac, 121 . 2.082785
to a u, 99.719 . 1.998779
To find B C.
A 8 radius. 10.000000
8 to cosine of C, 55° 30'. 9.753128
a c, 121 . 2.082785
to
B 0,68.535 . 1.835913
th hose, b c, being radius, then a b is the tangent, and a c6 secant of the angle C. Hence,
To find a b.
A 3 secant of c, 55° 30' . 10.246872
8 1° tangent of c, 55° 30' . 10.162866
A 13 Ac, 121 . 2.082785
4 b, 99.719 . 1.998779
As
To find b c.
„ secant of c, 55° 30' . 10.246872
8 1 0
radius . 10.000000
t ' SA c, 121 . 2.082785
B C, 68.535 .. 1.835913
" e perpendicular, a b, being radius, then bc becomesl n( j tan g e nt of the angle a, or the cotangent of the angle c ;
tliT 4 c becomes the secant of angle a, or cosecant of angle c.tQ ce,
^ To find a b.
1 °°secant of c, 55° 30' . 10.084006
is t ^cant ot c, oo“ oo . tu.uowuo
s 0 . radius . 10.000000
t 0 ’ s 4 c,121 . 2.082785
4 99.719 . 1.998779
^ To find b c.
is 8 . c °secant ofc, 55° 30'. 10.084006
So .° cotangent of c, 55° 30' . 9.837134
t 0 1 8 4 c,121 . 2.082785
c > 68.535 . 1.835913
^ General Rule for Gunter's Scale.
tl, at Xf end the compasses from the first term to the second ;ob 8 , r e *tent will reach from the third to the fourth term ;Y ' Ul g to take the line marked Num. for feet, yards,. H. 6 N
miles, &c., the line marked s. for sines of angles, and thatmarked T. for tangents. The radius is 90° of sines, and 45°of tangents.
Example 2.—In the triangle ABC, right-angled at b, letthe hypothenuse, A c, be 1045 feet, and the angle, a, 35° 56';what is the length of the base and perpendicular ?
Example 3. — A ship, from latitude 20° 30' north,sailed n.w. by n. 235 leagues ; what is her departure fromthe meridian, what her difference of latitude, and what thelatitude come to ?
Example 4.—Suppose one end of a rope, 350|- fathomslong, fixed at the top of an eminence, and the other endbrought down to the plane below, so that its direction makewith the plane an angle of 50° 40'; required the perpen-.dicular height of the eminence, and the space of the levelcovered by the rope.
Problem II.— Given the angles and one side, to find thehypothenuse and other side.
Example 1.—In the right-angled triangle ABC, right-angled at b, let the angle at A be 35° 30', and the side A b294 feet; required the base, b c, and the hypothenuse, a c.
Geometrically.
Make ab = 294 ; at the point a make an angle of 35°30*from the line of chords ; and from b raise the perpen-dicular b c. Measure b c and a c severally, by taking themin the compasses, and applying them to the scale from which,A b was taken ; then
By Calculation.
The hypothenuse, a c, being radius,
To find b c.
As cosine of A, 35° 30'. 9.910686
is to sine of A, 35° 30'... 9.763954
so is A b, 294 . 2.468347
to b c, 209.7 . 2.321615
To find A c.
As cosine of a, 35° 30' . 9.910686
is to radius . 10.000000
so is a b, 294 . 2.468347
to ac, 361.13 . 2.557661
The base being radius,
To find b c.
As cotangent of A, 35“ 30' . 10.146732
is to radius . 10.000000
so is a b, 294 . 2.468347
to bc, 209.7.. 2.321615
To find a c.
As cotangent of a, 35° 30' . 10.146732
is to cosecant of a, 35° 30'... 10.236046
so is A b, 294 ... 2.468347
to ac, 361.13 . 2.557661
The perpendicular, a b, being radius,
To find b c.
As radius... 10.000000
is to tangent of a, 35° 30' ... 9.853268
so is a b, 294 ... 2.468347
to bc, 209.7... 2.321165
To find a c.
As radius. 10.000000
is to secant of A, 35° 30'. 10.089314
so is a b, 294 . 2.468347
to AC, 361.13 . 2.557661