Tangenttilause on euklidisen geometrian perustulos.[1] Sen mukaan kolmiossa, jonka kaksi kulmaa ovat α {\displaystyle \alpha } ja β {\displaystyle \beta } ja näitä vastaavien sivujen pituudet ovat a {\displaystyle a} ja b {\displaystyle b} , on voimassa: a + b a − b = tan α + β 2 tan α − β 2 {\displaystyle {\frac {a+b}{a-b}}={\frac {\tan {\frac {\alpha +\beta }{2}}}{\tan {\frac {\alpha -\beta }{2}}}}} . Kolmio. Sinilauseen mukaan a sin α = b sin β . {\displaystyle {\frac {a}{\sin \alpha }}={\frac {b}{\sin \beta }}.} Olkoon d = a sin α = b sin β , {\displaystyle d={\frac {a}{\sin \alpha }}={\frac {b}{\sin \beta }},} jolloin a = d sin α ja b = d sin β . {\displaystyle a=d\sin \alpha {\text{ ja }}b=d\sin \beta .\,} Tällöin a − b a + b = d sin α − d sin β d sin α + d sin β = sin α − sin β sin α + sin β . {\displaystyle {\frac {a-b}{a+b}}={\frac {d\sin \alpha -d\sin \beta }{d\sin \alpha +d\sin \beta }}={\frac {\sin \alpha -\sin \beta }{\sin \alpha +\sin \beta }}.} Kun käytetään identiteettiä sin ( α ) ± sin ( β ) = 2 sin ( α ± β 2 ) cos ( α ∓ β 2 ) , {\displaystyle \sin(\alpha )\pm \sin(\beta )=2\sin \left({\frac {\alpha \pm \beta }{2}}\right)\cos \left({\frac {\alpha \mp \beta }{2}}\right),\;} saadaan a − b a + b = 2 sin 1 2 ( α − β ) cos 1 2 ( α + β ) 2 sin 1 2 ( α + β ) cos 1 2 ( α − β ) = tan [ 1 2 ( α − β ) ] tan [ 1 2 ( α + β ) ] . {\displaystyle {\frac {a-b}{a+b}}={\frac {2\sin {\tfrac {1}{2}}\left(\alpha -\beta \right)\cos {\tfrac {1}{2}}\left(\alpha +\beta \right)}{2\sin {\tfrac {1}{2}}\left(\alpha +\beta \right)\cos {\tfrac {1}{2}}\left(\alpha -\beta \right)}}={\frac {\tan[{\frac {1}{2}}(\alpha -\beta )]}{\tan[{\frac {1}{2}}(\alpha +\beta )]}}.} Sinilause Kosinilause Kotangenttilause [1]Thompson, Jan & Martinsson, Thomas: Matematiikan käsikirja, s. 376. Helsinki: Tammi, 1994. ISBN 951-31-0471-0. Tämä matematiikkaan liittyvä artikkeli on tynkä. Voit auttaa Wikipediaa laajentamalla artikkelia. Wikiwand in your browser!Seamless Wikipedia browsing. On steroids.Every time you click a link to Wikipedia, Wiktionary or Wikiquote in your browser's search results, it will show the modern Wikiwand interface.Wikiwand extension is a five stars, simple, with minimum permission required to keep your browsing private, safe and transparent.Wikiwand for ChromeWikiwand for EdgeWikiwand for Firefox
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