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TRIGONOMETRIC FORMUL S
<br />B B B
<br />c
<br />� c a' c a a
<br />b
<br />AC
<br />Cb CA b �
<br />{ Right Triangle Oblique Triangles
<br />Solution of Right 'triangles .
<br />a b a b c c
<br />For Angle A. sin=. ,cos= c ,tan= b , cot= a , sec = b, cosec'= a
<br />Given Required a
<br />a,b A,B,c ton A=b=cotB,c= az+bl=a 1+a2
<br />z
<br />a, c A, B, b sin A = a = cos B, b = \1(c+a) (c—a) = c J 1_L i .
<br />i c
<br />a
<br />A,.a B, b,'c B=90°—A,.b = acotA,c =sin A.
<br />A, b B, a, e B = 90°—A, a = b tan A, e = cos A.
<br />( A, c B, a, b B = 90°—A, a = c sin A, b = c cos A ,
<br />Solution of Oblique Triangles
<br />Given Required
<br />A, B, a b, c, C b — a sin B � C = 180°—(A + B), c = a sin C
<br />sin A sin A
<br />b sin A a sin C
<br />1 A, B, c, C sin B = a , C = 180'—(A + B), c = sin A
<br />a, b, C A, B, c A+B=180°— C, tan a (A—B)= (a—b) tan a +B)
<br />asinC a+b
<br />'
<br />sin A
<br />ABC s = 2 in2'A= b c ,
<br />I sin2B=�(s—aa(a c),C=180°—(A-+"B)
<br />1 a, b, c Area s = a i 2 c , area = s (s—a (s—b) (s—c)
<br />A, b, c Area area = b c sin A
<br />2
<br />a2 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 />cosine of the vertical angle. Thus: slope distance =319.4 ft.
<br />taoce Vert. angle=5° 101. From Table, Page IX. cos 5'10/=
<br />a�5 9959. Horizontal distance=319.4X.9959=318.09 ft.
<br />r��o4e "e Horizontal distance also = Slope distance minus slope
<br />Vel 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 50 101=.9959.1—.9959=.0041.
<br />319.4X.0041=1.31-319.4-1.31=318.09 ft.
<br />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 />_
<br />slope distance=302.6 ft. Horizontal distance=302.6— 14X14 —302.6-0.32=302.28 ft.
<br />2 X 302.6
<br />t MADE W u. e. N.
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<br />z S
<br />�
<br />3�
<br />� 7
<br />TRIGONOMETRIC FORMUL S
<br />B B B
<br />c
<br />� c a' c a a
<br />b
<br />AC
<br />Cb CA b �
<br />{ Right Triangle Oblique Triangles
<br />Solution of Right 'triangles .
<br />a b a b c c
<br />For Angle A. sin=. ,cos= c ,tan= b , cot= a , sec = b, cosec'= a
<br />Given Required a
<br />a,b A,B,c ton A=b=cotB,c= az+bl=a 1+a2
<br />z
<br />a, c A, B, b sin A = a = cos B, b = \1(c+a) (c—a) = c J 1_L i .
<br />i c
<br />a
<br />A,.a B, b,'c B=90°—A,.b = acotA,c =sin A.
<br />A, b B, a, e B = 90°—A, a = b tan A, e = cos A.
<br />( A, c B, a, b B = 90°—A, a = c sin A, b = c cos A ,
<br />Solution of Oblique Triangles
<br />Given Required
<br />A, B, a b, c, C b — a sin B � C = 180°—(A + B), c = a sin C
<br />sin A sin A
<br />b sin A a sin C
<br />1 A, B, c, C sin B = a , C = 180'—(A + B), c = sin A
<br />a, b, C A, B, c A+B=180°— C, tan a (A—B)= (a—b) tan a +B)
<br />asinC a+b
<br />'
<br />sin A
<br />ABC s = 2 in2'A= b c ,
<br />I sin2B=�(s—aa(a c),C=180°—(A-+"B)
<br />1 a, b, c Area s = a i 2 c , area = s (s—a (s—b) (s—c)
<br />A, b, c Area area = b c sin A
<br />2
<br />a2 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 />cosine of the vertical angle. Thus: slope distance =319.4 ft.
<br />taoce Vert. angle=5° 101. From Table, Page IX. cos 5'10/=
<br />a�5 9959. Horizontal distance=319.4X.9959=318.09 ft.
<br />r��o4e "e Horizontal distance also = Slope distance minus slope
<br />Vel 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 50 101=.9959.1—.9959=.0041.
<br />319.4X.0041=1.31-319.4-1.31=318.09 ft.
<br />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 />_
<br />slope distance=302.6 ft. Horizontal distance=302.6— 14X14 —302.6-0.32=302.28 ft.
<br />2 X 302.6
<br />t MADE W u. e. N.
<br />L
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