Molveide Formulas

Triangle on the plane.

Molveide formulas are trigonometric dependencies expressing the relationship between the lengths of the sides and the values of the angles at the vertices of a certain triangle, discovered by K. B. Mollveide .

Content

Description

Molveide formulas have the following form:

{\frac {a+b}{c}}={\frac {\operatorname {cos} \;{\frac {A-B}{2}}}{\operatorname {sin} \;{\frac {C}{2}}}};

{\ displaystyle {\ frac {a + b} {c}} = {\ frac {\ operatorname {cos} \; {\ frac {AB} {2}}} {\ operatorname {sin} \; {\ frac { C} {2}}}};}

{\frac {a-b}{c}}={\frac {\operatorname {sin} \;{\frac {A-B}{2}}}{\operatorname {cos} \;{\frac {C}{2}}}},

{\ displaystyle {\ frac {ab} {c}} = {\ frac {\ operatorname {sin} \; {\ frac {AB} {2}}} {\ operatorname {cos} \; {\ frac {C} {2}}}},}

where A , B , C are the values of the angles at the corresponding vertices of the triangle and a , b , c are the lengths of the sides, respectively, between the vertices B and C , C and A , A and B. The formulas are named after the German mathematician Karl Molweide . Molveide formulas are convenient to use when solving a triangle on two sides and the angle between them ^[1] and at two angles and the side adjacent to them. Similar relations in spherical trigonometry are called Delambre formulas ^[1] .

Application

Separating the right and left sides of the last formulas separately, we immediately obtain the tangent theorem

Notes

↑ ¹ ² Stepanov N.N. Spherical trigonometry. - M. - L .: OGIZ , 1948 .-- 154 p.

Literature

O. V. Manturov , Yu. K. Solntsev , Yu. I. Sorkin , N. G. Fedin . Explanatory Dictionary of Mathematical Terms, Moscow: Enlightenment, 1965.
Trigonometric Identities
Trigonometric functions

[St-1] ¹ ² Stepanov N.N. Spherical trigonometry. - M. - L .: OGIZ , 1948 .-- 154 p.