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Half-side formula
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<h2 id="half-tangent-variant">Half-tangent variant</h2> <p>The same relationships can be written as rational equations of half-tangents (tangents of half-angles). If t a = tan 1 2 a , {\displaystyle t_{a}=\tan {\tfrac {1}{2}}a,} t b = tan 1 2 b , {\displaystyle t_{b}=\tan {\tfrac {1}{2}}b,} t c = tan 1 2 c , {\displaystyle t_{c}=\tan {\tfrac {1}{2}}c,} t A = tan 1 2 A , {\displaystyle t_{A}=\tan {\tfrac {1}{2}}A,} t B = tan 1 2 B , {\displaystyle t_{B}=\tan {\tfrac {1}{2}}B,} and t C = tan 1 2 C , {\displaystyle t_{C}=\tan {\tfrac {1}{2}}C,} then the half-side formula is equivalent to: </p><p> t a 2 = ( t B t C + t C t A + t A t B − 1 ) ( − t B t C + t C t A + t A t B + 1 ) ( t B t C − t C t A + t A t B + 1 ) ( t B t C + t C t A − t A t B + 1 ) . {\displaystyle {\begin{aligned}t_{a}^{2}&={\frac {{\bigl (}t_{B}t_{C}+t_{C}t_{A}+t_{A}t_{B}-1{\bigr )}{\bigl (}{-t_{B}t_{C}+t_{C}t_{A}+t_{A}t_{B}+1}{\bigr )}}{{\bigl (}t_{B}t_{C}-t_{C}t_{A}+t_{A}t_{B}+1{\bigr )}{\bigl (}t_{B}t_{C}+t_{C}t_{A}-t_{A}t_{B}+1{\bigr )}}}.\end{aligned}}} </p><p>and the half-angle formula is equivalent to: </p><p> t A 2 = ( t a − t b + t c + t a t b t c ) ( t a + t b − t c + t a t b t c ) ( t a + t b + t c − t a t b t c ) ( − t a + t b + t c + t a t b t c ) . {\displaystyle {\begin{aligned}t_{A}^{2}&={\frac {{\bigl (}t_{a}-t_{b}+t_{c}+t_{a}t_{b}t_{c}{\bigr )}{\bigl (}t_{a}+t_{b}-t_{c}+t_{a}t_{b}t_{c}{\bigr )}}{{\bigl (}t_{a}+t_{b}+t_{c}-t_{a}t_{b}t_{c}{\bigr )}{\bigl (}{-t_{a}+t_{b}+t_{c}+t_{a}t_{b}t_{c}}{\bigr )}}}.\end{aligned}}} </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Spherical_law_of_cosines/DYT3Er58">Spherical law of cosines</a></li> <li><a href="/facts/Law_of_haversines/NGJlq4q7">Law of haversines</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Bronshtein, I. N.; Semendyayev, K. A.; Musiol, Gerhard; Mühlig, Heiner (2007), Handbook of Mathematics, Springer, p. 165, ISBN 9783540721222[1] <a href="9783540721222" target="_blank">9783540721222</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Nelson, David (2008), The Penguin Dictionary of Mathematics (4th ed.), Penguin UK, p. 529, ISBN 9780141920870. <a href="9780141920870" target="_blank">9780141920870</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> </ol>