To find BC.

To find AC.

Astang.ang.C 59° 20' 10.22697 | As tan. C 59° 20'

is to AB 300

is to AB

300

So fec. C 59° 20' - 10.29239

So rad. 90°

Variety 3. Making AC rad. BC becomes the fine of angle A, and AB fine angle C. Hence the following proportions.

Ex. 2. In the right angled triangle ABC, right angled at B, fuppofe BC-4876 equal parts, angle A 53° 3', and angle C 36° 57'. Required AB; AC. Fig 52. plate 4.

Conftruction. Draw AB, upon B erect the perpendicular BC 4876, and at C draw CA, making an angle of 36° 57' with BC; then shall angle A be 53° 3', AB 3668, and AC 6101 equal parts.

Variety 1. Making AB rad. BC becomes the tangent, and AC the fecant of angle A. Hence arife the following propor

tions.

To find AB.

As tan. ang. A 53°3′ 10.12367 | As tan. A 53° 3'

to BC 4876 So rad. 90

3.68806

10.00000

3.56439

To find AC.

Variety 2. Making BC rad. AB becomes the tangent, and AC the fecant of angle C. Hence arife the following propor

Variety 3. Making AC rad. AB becomes the fine of the angle at C, and BC the fine of the angle at A. Whence arise the following proportions.

The angles and the hypothenuse being given to find the legs.

In the triangle ABC, right angled at B. suppose AC 568 cqual parts, angle A 39° 14′, and angle C 50° 46'. Required AB, BC. Fig. 44. plate 3.

Variety 1. Making AB radius, BC becomes the tangent, and AG the fecant of A. Whence the following proportions.

Is to AC 568,

10.11094 As fec. A 39° 14'
2.75435 Is to AC 568

10.11004

So is tan. A 39° 14' 9.91198 So is rad. 90

Variety 2. Making BC rad. AB becomes the tangent, and AC the fecant of the angle C. Whence the following propor

tions.

To find BC.

To find AB.

As fec. C 50° 46' - 10.19895 | As fec. C 50° 46' - 10.19895

2.75435

10.00000

2.75435

is to AC 568 So is tan. C 50° 46' 10.08802

is to AC 568 So is rad. 90

Variety 3. Making AC rad. BC becomes the fine of angle A, and AB the fine of the angle at C. Whence the following proportions.

So is fine A 39° 14' 9.80105 So is fine C 50°

46

9.88906

Two fedes and the right angle given, to find the acute angles, and the third fide. Fig. 46. plate 3.

In the triangle ABC, right angled at B, suppose the hypothe nufe AC 150, and the leg. CB 69. Required the angles A and C and the leg. BA.

Variety 1. Making AC rad. then BC becomes the fine of the angle at A. Whence, the following proportion.

To find angle A.

9.66276

To find angle C.

2.17609 | Since the two angles of a right 10.00000 angled triangle, are comple 1.83885 ments of each other, angle C may be found, (by fubtracting angle A=27° 23′ from 90°,) to be 62° 37'

Variety 2. Making BC rad. then AC becomes the fecant of the angle at C. Whence the following proportion.

Te find angle C.

To find angle A.

1.83885 If from 90° you fubtract 62° 10.00000 27', the remainder 27° 23′ will 2.17609 give angle A.

To fec. C. 62° 37' 10 33724

Now there are other three varieties to find AB.

When an angle is required, the length of a line is made the first and third terms, also a fide that is neither given nor required, cannot be admitted into the proportion, or made radius..