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  Basic Formulas
  Sum and Difference Formulas
  Double & Triple Angle Formulas
  Half Angle Formulas
  Product to Sum Formulas
  Sum to Product Formulas
  All Formulas

   Final thoughts

Sum and Difference Formulas:

Sum and Difference Formulas (10028 bytes)

There are four possible products of two factors that can be formed from cos_alph.gif (998 bytes), sin_alph.gif (999 bytes), cos_beta.gif (1013 bytes), and sin_beta.gif (1011 bytes). Namely: 3_02.gif (1206 bytes), 3_03.gif (1227 bytes), 3_04.gif (1229 bytes), 3_05.gif (1227 bytes). Note their distribution in the first four formulas. The right hand sides of (1) and (2) consist of the algebraic sum of 3_02.gif (1206 bytes) and 3_03.gif (1227 bytes), while (3) and (4) consist of the mixed products. Note the distribution of signs below.

3_06.gif (3902 bytes)

In formulas 1 and 2, for cosine the signs on each side of the formulas are "opposite". Be aware of the order of the subtraction: the product of cosine minus the product of sine.

3_107.gif (3946 bytes)

In formulas 3 and 4, for sine the signs on both sides are the same. Be aware of the subtraction in formula 4: in front of the product is negative, when the product contains the factor sin_beta.gif (1011 bytes), where beta.gif (897 bytes) is from the term of al_m_bet.gif (996 bytes).

3_108.gif (3889 bytes)

Formulas 5 and 6, tangents have the same sign in the numerator of both sides. The numerator is also what you get if you multiply tangent by 3_109.gif (1142 bytes) (i.e. 3_110.gif (1927 bytes). Notice, this is not actually how it works). The denominator is 3_111.gif (1379 bytes), the sign being the opposite of the numerator.


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