LAWS OF SINE, COSINE, AND TANGENT
In any triangle
Law of Sines
= =
Law of Cosines
a
2
= b
2
+ c
2
– 2bc cos A b
2
= a
2
+ c
2
– 2ac cos B c
2
= a
2
+ b
2
– 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 all positive (+)
Quad II sin (+), cos (-), tan (-)
Quad III sin (-), cos (-), tan (+)
Quad IV sin (-), cos (+), tan (-)
TRIGONOMETRIC ELEVATION COMPUTATIONS
ζ
1
= mean observed ZD S = geodetic distances
sin 1″ = 0.00000485 T = slope distance
Reduction of Reciprocal Zenith Distance Observations
Reduction in seconds = −
Reciprocal Observations
h
2
– h
1
= T sin ½(ZD
2
− ZD
1
) or h
2
− h
1
= S tan ½(ZD
2
− ZD
1
)
Nonreciprocal Observations
h
2
−
h
1
= T sin (90° − ζ
1
+ k) or h
2
− h
1
= S tan (90° − ζ
1
+ k)
Legend:
C C factor cm centimeter(s) cos cosine
cot cotangent csc cosecant FM field manual
FS far sight ft foot; feet GTA graphic training aid
in inch(es) kg kilogram(s) km kilometer(s)
lb pound(s) LEC linear error of closure m meter(s)
mi mile(s) mm millimeter(s) sec secant
SIF stadia-interval factor sin sine SLC sea level coefficient
t grid azimuth tan tangent ZD zenith distance
*GTA 05-02-029
1 April 05
Conversion
Factors
and
Common
Formulas
Purpose: Use this GTA as a guide for making common conversions.
See FM 3-34.331 for more information.
DISTRIBUTION: Installation Training Support Centers (TSCs).
DISTRIBUTION RESTRICTION: Approved for public release;
distribution is unlimited.
Headquarters, Department of the Army
*This publication supersedes GTA 05-02-029, August 1987.
B
a
C
b
A
c
IV
I
III
II
0°
90°
270°
180°
(HI - HT) sin mean ZD
S sin 1″
4
sin A
a
sin B
b
sin C
c
