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Sinusrelation Cosinusrelation

Sinusrelation Cosinusrelation. Beviser. Forudsætninger: Definition af Sinus (i enhedscirklen) Definition af Cosinus (i enhedscirklen) Sin og Cos i den retvinklede trekant. hyp. hyp ∙ sin(v). v. hyp ∙ cos (v). c. a. A. C. sin(A). 1. sin(A). A.

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Sinusrelation Cosinusrelation

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  1. SinusrelationCosinusrelation Beviser

  2. Forudsætninger: Definition af Sinus (i enhedscirklen) Definition af Cosinus (i enhedscirklen) Sin og Cos i den retvinklede trekant hyp hyp∙sin(v) v hyp∙cos(v)

  3. c a A C sin(A) 1 sin(A) A Sinusrelationen i en retvinklet trekant Da trekanterne er ensvinklede får vi samme forhold mellem de to lodrette sider, som mellem de to skrå sider: Bruger vi 1 = sin(90°) = sin(C) fås: (Eller på hovedet):

  4. Sinusrelation i ”skævvinklet” trekant P r q R Q p r q h Q R

  5. Vi bruger om retvinklede trekanter: h h r r∙sin(Q) q q∙sin(R) hyp∙sin(v) hyp Q R v

  6. Cosinusrelation i ”skævvinklet” trekant P r q R Q p r q h Q p

  7. P r q r q h h = r·cos(Q) R Q Q p y x Pyth: h2= r2 – x2 h2= q2– y2 h2= h2 r2 – x2 = q2– y2 q2– y2 = r2 – x2 q2 = r2 – x2+ y2 q2 = r2 – x2+ p2+ x2– 2∙p∙x q2 = r2 + p2 – 2∙p∙x q2 = r2 + p2 – 2∙p∙ r∙cos(Q) Da x+y = p fås y = p – x, hvoraf y2 = (p – x)2 = (p – x)·(p – x) = p2 – p·x – x·p + x2 = p2 + x2 – 2·p·x hyp v hyp∙cos(v) Vi bruger om retvinklede trekanter:

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