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I
<br />TRIGONOMETRIC FORMULfE
<br />B B B
<br />e a
<br />- c a c a
<br />A b A V C
<br />e C �bC A
<br />Right Triangle Oblique Triangles
<br />1 Solution of Right Triangles
<br />For Angle A. sin = a ,cos = b ,tan = a b
<br />cot = a ,sec = �, cosec =
<br />b.
<br />Given Required
<br />a,b A, ,c on b=cotB,c= a -}- �=a 1 { R a
<br />- _ a
<br />a, c A, B, b sin A = = cos B, b c Y 1— a2
<br />V O
<br />A, a B, b, c B=90°=A, b= a cotA, c= a
<br />sin A.
<br />A, b B, a, c B = 90°—A, a = b tan A, c = b
<br />' cos A.
<br />A, c' :A a, b B =900—A. a = c sin A, b = c cos A,
<br />Solution of Oblique Triangles
<br />Given Required a sin B
<br />A, B, a b, c, C b = , C = 180°—(A + B), c = a sin C
<br />sin A sin A
<br />b sin Aa sin C
<br />A, a, b B, c, C+ 'sin B = a C ='180'—(A +. B) , c = sin A
<br />a
<br />a, b, C A, B,.0- A+B.180°—C, tan 1,(A—B)= a m + (�-
<br />(A +B) .
<br />a sin C'
<br />sin A
<br />a, b, a A, B, C a=a-f-�+e,sin zA= ) a b(a_c ,
<br />sin 1B=,C=180°---(A+B)
<br />v ac
<br />a+b+c
<br />a, b, a Area S= 2 ,area =s(s—a s—
<br />bcsinA
<br />A, b, c Area area =
<br />2
<br />a"- sin B sin C
<br />A, B, C, a Area area = 2 sin A
<br />REDUCTION TO HORIZONTAL _
<br />Horizontal distance= Slope distance multiplied by the
<br />0 cosine of the vertical angle. Thus: slope distance =3L9.4ft.
<br />e alstorc y Vert. angle =50 101. From Table, Page IX, cos b° 10/= ft
<br />9959. Horizontal distance=319.4X.9959=818.09 .
<br />SX09 rgte Horizontal distance also=Slope distance minus slope+
<br />Ve a distance times (1—cosine of vertical angle). With the
<br />same figures as in the preceding example, the follow -
<br />Horizontal distance ing result is obtained. Cosine 5'10'=.0959. 1—.0959=.0041.
<br />319.4X.0041=1.31.319.4-1.31=318.09 ft.
<br />i When the rise is known, the horizontal distance is approximately:—the slope dist-
<br />ance less the square of the rise divided by twice the slope distance. Thus: rise =14 ft.,
<br />slope distance=302.8 ft. Horizontal distance=3026— 14 X 14 =3026-0.32=302.28 ft.
<br />2 X 302.6
<br />a
<br />• ` W MADE IN V.$,A.
<br />U
<br />�i
<br />1
<br />I
<br />TRIGONOMETRIC FORMULfE
<br />B B B
<br />e a
<br />- c a c a
<br />A b A V C
<br />e C �bC A
<br />Right Triangle Oblique Triangles
<br />1 Solution of Right Triangles
<br />For Angle A. sin = a ,cos = b ,tan = a b
<br />cot = a ,sec = �, cosec =
<br />b.
<br />Given Required
<br />a,b A, ,c on b=cotB,c= a -}- �=a 1 { R a
<br />- _ a
<br />a, c A, B, b sin A = = cos B, b c Y 1— a2
<br />V O
<br />A, a B, b, c B=90°=A, b= a cotA, c= a
<br />sin A.
<br />A, b B, a, c B = 90°—A, a = b tan A, c = b
<br />' cos A.
<br />A, c' :A a, b B =900—A. a = c sin A, b = c cos A,
<br />Solution of Oblique Triangles
<br />Given Required a sin B
<br />A, B, a b, c, C b = , C = 180°—(A + B), c = a sin C
<br />sin A sin A
<br />b sin Aa sin C
<br />A, a, b B, c, C+ 'sin B = a C ='180'—(A +. B) , c = sin A
<br />a
<br />a, b, C A, B,.0- A+B.180°—C, tan 1,(A—B)= a m + (�-
<br />(A +B) .
<br />a sin C'
<br />sin A
<br />a, b, a A, B, C a=a-f-�+e,sin zA= ) a b(a_c ,
<br />sin 1B=,C=180°---(A+B)
<br />v ac
<br />a+b+c
<br />a, b, a Area S= 2 ,area =s(s—a s—
<br />bcsinA
<br />A, b, c Area area =
<br />2
<br />a"- sin B sin C
<br />A, B, C, a Area area = 2 sin A
<br />REDUCTION TO HORIZONTAL _
<br />Horizontal distance= Slope distance multiplied by the
<br />0 cosine of the vertical angle. Thus: slope distance =3L9.4ft.
<br />e alstorc y Vert. angle =50 101. From Table, Page IX, cos b° 10/= ft
<br />9959. Horizontal distance=319.4X.9959=818.09 .
<br />SX09 rgte Horizontal distance also=Slope distance minus slope+
<br />Ve a distance times (1—cosine of vertical angle). With the
<br />same figures as in the preceding example, the follow -
<br />Horizontal distance ing result is obtained. Cosine 5'10'=.0959. 1—.0959=.0041.
<br />319.4X.0041=1.31.319.4-1.31=318.09 ft.
<br />i When the rise is known, the horizontal distance is approximately:—the slope dist-
<br />ance less the square of the rise divided by twice the slope distance. Thus: rise =14 ft.,
<br />slope distance=302.8 ft. Horizontal distance=3026— 14 X 14 =3026-0.32=302.28 ft.
<br />2 X 302.6
<br />a
<br />• ` W MADE IN V.$,A.
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