A Treatise on Plane and Spherical TrigonometryH. Perkins, 1852 |
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Page 39
... second member by ( 103 ) and ( 134 ) ; - 4 sin v sin ( v — x ) sin ( v — y ) sin ( v — z ) - In the same manner we find -- - 2 cos x cos y cos z- cos y cos z + 1 cos2 x ( 173 ) 4 cos v cos ( v - x ) cos ( v — y ) cos ( v — z ) = 2 cos x ...
... second member by ( 103 ) and ( 134 ) ; - 4 sin v sin ( v — x ) sin ( v — y ) sin ( v — z ) - In the same manner we find -- - 2 cos x cos y cos z- cos y cos z + 1 cos2 x ( 173 ) 4 cos v cos ( v - x ) cos ( v — y ) cos ( v — z ) = 2 cos x ...
Page 40
... member of ( 170 ) being thus reduced to zero , the numerator of the second number must be zero , or tan x + tan y + tan z = tan x tan y tan z ( 181 ) an equation , it must be remembered , that is true only under the condition ( 180 ) ...
... member of ( 170 ) being thus reduced to zero , the numerator of the second number must be zero , or tan x + tan y + tan z = tan x tan y tan z ( 181 ) an equation , it must be remembered , that is true only under the condition ( 180 ) ...
Page 59
... members from unity ; then 2 bcb2c2 + a2 a2 — ( b — c ) 2 1 - cos A = 2 bc 2 bc But , by ( 139 ) , making x = A , we have } 1 - cos A = 2 sin2 1⁄2 A Also , the numerator of the second member being the difference of two squares may be ...
... members from unity ; then 2 bcb2c2 + a2 a2 — ( b — c ) 2 1 - cos A = 2 bc 2 bc But , by ( 139 ) , making x = A , we have } 1 - cos A = 2 sin2 1⁄2 A Also , the numerator of the second member being the difference of two squares may be ...
Page 71
... second member of ( 266 ) is always < 1 . EXAMPLE . Given a 062387 , b : = a = 062387 b = .023475 a + b = · 085862 C = 55 ° 16 ' 023475 , C 110 ° 32 ' ; ( same as Ex . 1 , Art . 138 ) By ( 266 ) and ( 267 ) . log ab log 8.7950941 log ...
... second member of ( 266 ) is always < 1 . EXAMPLE . Given a 062387 , b : = a = 062387 b = .023475 a + b = · 085862 C = 55 ° 16 ' 023475 , C 110 ° 32 ' ; ( same as Ex . 1 , Art . 138 ) By ( 266 ) and ( 267 ) . log ab log 8.7950941 log ...
Page 101
... two functions , which may be applied to either to obtain the other . 187. To find the increment of the sine or cosine ... second member will be negative so long as x < 180 ° , and therefore the increment of the cosine is negative ; that ...
... two functions , which may be applied to either to obtain the other . 187. To find the increment of the sine or cosine ... second member will be negative so long as x < 180 ° , and therefore the increment of the cosine is negative ; that ...
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Common terms and phrases
applied B+AB becomes Bowditch's Rules C+AC C₁ computation constant cos A cos cos A sin cosc cosec cosine cosm cot A cot deduce denote difference differential divided employed equal equations EXAMPLE expressed factors formulæ gives hypotenuse increments less than 180 log cot logarithms Napier's Rules negative obtain perpendicular plane triangle polar triangle positive preceding article quadrant quotient radius reduced right angle right triangles secant second member simple angle sin b cos sin b sin sin x sin² sin² ½ sine sine and cosine solution solve the triangle spherical triangle SPHERICAL TRIGONOMETRY tables tan-¹ tan² tana tangent theorem Trig trigonometric functions whence Δα
Popular passages
Page 151 - Spherical Triangle the cosine of any side is equal to the product of the cosines of the other two sides, plus the product of the sines of those sides into the cosine of their included angle ; that is, (1) cos a = cos b...
Page 58 - THE SUM OF THE OPPOSITE ANGLES', To THE TANGENT OF HALF THEIR DIFFERENCE. Thus, the sum of AB and AC, (Fig. 25.) is to their difference ; as the tangent of half the sum of the angles ACB and ABC, to the tangent of half their difference.
Page 58 - In any plane triangle, the sum of any two sides is to their difference, as the tangent of half the sum of the opposite angles is to the tangent of half their difference.
Page 15 - The sum of the two acute angles of a right triangle is equal to one right angle, or 90°.
Page 34 - I sin y \2 / \2 / = sin x cos y + cos x sin y, sin (a; — y) = sin (x + (— y)) = sin a; cos (— y) + cos a; sin (— y) = sin x cos y — cos x sin y, tan (x + y) = sin (x + y) sin x cos y + cos x...
Page 64 - As the sine of the angle opposite the given side, is to the sine of the angle opposite the required side ; so is the given side to the required side.
Page 65 - The side opposite the given angle is to the side opposite the required angle as the sine of the given angle is to the Bine of the required angle.
Page 179 - ... the sign of cos A, is the same as that of cos a, that is, A and a are in the same quadrant.
Page 150 - The law of sines states that in any spherical triangle the sines of the sides are proportional to the sines of their opposite angles: sin a _ sin b __ sin c _ sin A sin B sin C...
Page 244 - If the sides of a triangle are very small compared with the radius of the sphere and a plane triangle be formed whose sides are equal to those of the spherical triangle...