Army Basic Math-convers.factors.comm.formula

LAWS OF SINE, COSINE, AND TANGENT B

In any triangle

c

a

A

C

b

Law of Sines sin A a

=

sin B b

=

sin C c

*GTA 05-02-029 1 April 05

Law of Cosines a2 = b2 + c2 – 2bc cos A

b2 = a2 + c2 – 2ac cos B

c2 = a2 + b2 – 2ab cos C

Law of Tangents a + b = tan[½(A + B)] a - b = tan[½(A - B)]

ALGEBRAIC SIGNS OF TRIG FUNCTIONS BY QUADRANT Quad I Quad II Quad III Quad IV

all positive (+) sin (+), cos (-), tan (-) sin (-), cos (-), tan (+) sin (-), cos (+), tan (-)

0° IV

I

III

II

90°

270°

180°

TRIGONOMETRIC ELEVATION COMPUTATIONS ζ1 = mean observed ZD sin 1″ = 0.00000485

S = geodetic distances T = slope distance

Conversion Factors and Common Formulas

Reduction of Reciprocal Zenith Distance Observations Reduction in seconds = −

(HI - HT) sin mean ZD S sin 1″

Reciprocal Observations h2 – h1 = T sin ½(ZD2 − ZD1)

or

h2 − h1 = S tan ½(ZD2 − ZD1)

DISTRIBUTION: Installation Training Support Centers (TSCs).

Nonreciprocal Observations h2 − h1 = T sin (90° − ζ1 + k)

or

h2 − h1 = S tan (90° − ζ1 + k)

Legend: C cot FS in lb mi SIF t

C factor cotangent far sight inch(es) pound(s) mile(s) stadia-interval factor grid azimuth

cm csc ft kg LEC mm sin tan

centimeter(s) cosecant foot; feet kilogram(s) linear error of closure millimeter(s) sine tangent 4

Purpose: Use this GTA as a guide for making common conversions. See FM 3-34.331 for more information.

cos FM GTA km m sec SLC ZD

cosine field manual graphic training aid kilometer(s) meter(s) secant sea level coefficient zenith distance

DISTRIBUTION RESTRICTION: Approved for public release; distribution is unlimited.

Headquarters, Department of the Army *This publication supersedes GTA 05-02-029, August 1987.

CONVERSION FACTORS

LEVELING ("C" FACTOR) C factor =

Length 1 mm 10 mm 100 mm 1,000 mm 10 m 100 m 1,000 m

= 0.100 cm = 1.000 cm = 10.000 cm = 100.000 cm = 0.010 km = 0.100 km = 1.000 km

= 0.001 m = 0.010 m = 0.100 m = 1.000 m

1 cm 1 in 1m 1 ft 1 km 1 mi

= 0.3937 in = 2.54 cm = 39.37 in = 3.28083 ft = 30.48 cm = 0.30480 m = 0.62137 mi = 1609.344 m = 1.60930 km

π

1 degree = 180 radians

1 radian =

180 π

degrees

°C = 5/9(°F - 32)

°F = (9/5 °C) + 32 Weight

1 lb = 0.4536 kg

1 kg = 2.2050 lb

B c

In any right triangle

a

A

Cosine =

adjacent leg hypotenuse

Tangent =

opposite leg adjacent leg

Cosecant =

1 sine

Secant =

Cotangent =

b

Sin A =

a c

Cos A =

b c

Tan A =

a b

Bearing (β) = tan ∆N if if if if

C

K = Ko [1 + (XVIII) q2 + 0.00003q4] Ko = 0.9996 q = 0.000001 x E′

K = K x SLC mean elevation

SLC = 1 – mean radius of the earth Mean radius of the earth = 6,372,000 m (20,906,000 ft) =

∆N = grid distance x cosine of azimuth

Sec A = b/c =

c b

∆E = grid distance x sine of azimuth

1 = a/c

1

Cot A = a/b = a = √c2 - b2

2

∆N + ∆N ∆N ∆N +

TRAVERSE COMPUTATIONS

2 2 LEC = √error ∆N + error ∆E

b a

Ratio of closure = 1: c = √a2 + b2

∆E + ∆E + ∆E ∆E -

K = scale factor (horizontal distance)

1

1 tangent

∆E

Distance = √ΔN2 +ΔE2

c a

Csc A

1 cosine

C ±0.003 ±0.007 ±0.010

K = scale factor (geodetic distance)

COMMON FORMULAS

opposite leg hypotenuse

SIF 1:100 1:200 1:333

Correct rod reading = C x last FS interval + last FS mean reading

t = grid azimuth t=β t = 180° - β t = 180° + β t = 360° - β

Temperature

Sine =

Maximum allowable C factor:

BASIC COMPUTATIONS

Degrees 1 degree = 17.777777 mils

near rod reading – far rod reading far intervals – near intervals

b = √c2 - a2

length (m) LEC

3