Cosab Formula

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It arises from the law of cosines and the distance formula.

Cosab formula. Finding cosa b a very similar construction finds the formula for the cosine of an angle made with two angles added together. All you have to do is a couple of substitutions and a multiply. Using the same construction 1 notice that the adjacent side is the full base line for cos a with part of it subtracted at the right.

Sin cos formulas are based on sides of the right angled triangle. Cosa b cosacosb sinasinb. Sum and difference formula sina b sin acosbcos asinbsina b sin acosb cos asinbcosa b cos acosb sin asinbcosa b cos acosbsin asinbtana b tan atanb 1 tan atanb tana b tan a tanb 1tan atanb double angle formula.

In just a few short steps the formulas for cosa b and sina b flow right from equation 47 eulers equation for e i xno more need to memorize which one has the minus sign and how all the sines and cosines fit on the right hand side. The third formula shown is the result of solving for a in the quadratic equation a 2 2ab cos g b 2 c 2 0. Sine of angle is equal to the ratio of opposite side and hypotenuse whereas cosine of an angle is equal to ratio of adjacent side and hypotenuse.

This equation can have 2 1 or 0 positive solutions corresponding to the number of possible triangles given the data. This video shows the formula for deriving the cosine of a sum of two angles.

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$\sin \frac{A}{2} = \sqrt{\dfrac{(s- b)(s- c)}{bc}} ; \, \, \, \cos \frac{A}{2} = \sqrt{\dfrac{s(s- a)}{bc}} \\ \sin \frac{B}{2} = \sqrt{\dfrac{(s- c)(s- a)}{ca} } ; \, \, \, \cos \frac{B}{2} = \sqrt{\dfrac{s(s- b)}{ca}}\\ \sin \frac{C}{2} = \sqrt{\dfrac{(s- c)(s- b)}{ab}} ; \, \, \, \cos \frac{C}{2} = \sqrt{\dfrac{s(s- c)}{ab}}\\ \tan \frac{A}{2} = \sqrt{\dfrac{s- b) (s- c)}{s(s- a)}} ; \, \, \, \tan \frac{B}{2} = \sqrt{\dfrac{(s- c)(s- a)}{s(s- b)}} \\ \tan \frac{C}{2} = \sqrt{\dfrac{(s- a)(s- b)}{s(s- c)}}$

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$\sin A \cos B + \cos A \sin B = \sin ( A + B ) \cdots{} i \\ \\ \sin A \cos B - \cos A \sin B = \sin ( A - B ) \cdots{} ii \\ \\ \cos A \cos B - \sin A \sin B = \cos ( A + B ) \cdots{} iii \\ \\ \cos A \cos B+ \sin A \sin B = \cos ( A - B ) \cdots{} iv$

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