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DISTANCES FROM SIDE STAKES FOR CROSS-SECTIONING Roadway of any Width. Sid. Slopes 1% to 1.

I. tee nintre twine opposite 7 under 'Tut or FIII- and tinder .3 reed 11 0, the diatenre out tr.. li. . '0 ... at ... Also, oPPottite 1 i under ••,-,d or Fin.. •nd tlnder I reed to 7 the

dinette. out nom toe •Ide Make •t richt Slope Stake ‘.., / Side Stake ',..."! CP Li ___ Grade Center Stake ..,.. Sada Stake' \ -.r. 165 lope Stake .. 3 . 0 .1 .2 I .3 .4 I .5 . . 7 .8 . -..- ,.. Distance out from Side. or Shoulder Stake

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7

TRIGONOMETRIC FORMULA:

A '

b C 0 C

Right Triangle Oblique Triangles Solution of Right Triangles

a b a b e e

For Angle A. sin t.• -; , cos= , tan =

i , cot =

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E ,

cosec .. -is Given a, b a, c A, a A, b A, Given A, B, a A, a, b a, b, C a, b, c a bi tan A = -b = cot B , = a' + = a Ail ± a-. sin A = = cos 11,b V (c+a)(c-a) a 1/ 1 - ct-1 B=90° -A, b a cot A, o= a

sin A. B, a, a B=90°-A,a b tan A, a=

cos A. B, a, b B=90°-A, a = sin A,b= e cos A,

Solution of Oblique Triangles Required

a sin 11 a sin C b, c,

Cb - sin A, C = 180°-(A B), o sin A

b sin A a sin C B, e, C sin B ,C 180°-(A -f- B), •=. a sin A A, B, c A +B=180°- C, tan A. B, C a, b, e Area A, b, c Area A, B,C, a Area

sNee

Required A, B ,e A, B, b B, b, c a sin C = sin A - rs-t-b+c,sin IA - , Ica- R 2 b ' sin B - AP 4)(8-C) C- 180°-4 A 4-B) a c + b+c

= a area = Vs(a-a) (a-8) (8-c) 2 b c sin A reel /UCH 2 a' sin B sin C 2 sin A REDUCTION TO HORIZONTAL

Horizontal distance= Slope distance multiplied by the cosine of the vertical angle. Thus: slope distance 319.4 ft. Vert. angle 5" 10'. From Table. Page IX. cos 5 10'= .9959. Horizontal distance=319.4 X.9959 318.09 ft. Horizontal distance also-- Slope distance minus slope distance times (1 --cosine of vertical angle). With the same figures as in the preceding example, the follow-Horizontal distance ing result is obtained. Cosine 5° 10'- .9959. 1 -.9959=.0041.

319.4 ,(.0041 =1.31. 319.4-1.31=318.09 ft.

When the rise is known, the horizontal distance Is epprozimately:- the slope dist-ance less the square of the rise divided by twice the slope distdist-ance. Thus: rise =14 ft.. slope distance- 302.8 ft. llorizontal distance-302.8 14 )" 14 3,2.6-0.32 302.28

2 X 302.8

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