In ΔABC is \(\boldsymbol{\frac{c\,-\,a.\cos B}{a.\sin B} }\) gelijk aan

(A,B,C zijn de grootten van de hoeken
tegenover de zijden met lengten a,b,c) 		A.   sin A
B.   cos A
C.   tan A
D.   cot A
E.   sec A
                 

[ 4-7972 - op net sinds 26.8.15-()-4.12.2023 ]

Translation in   E N G L I S H

In ΔABC is \(\boldsymbol{\frac{c\,-\,a.\cos B}{a.\sin B} }\) equal to

(A,B,C are the measures of the angles oposite the sides a, b and c.) 		A.   sin A
B.   cos A
C.   tan A
D.   cot A
E.   sec A

Oplossing - Solution

Wegens de sinusregel   \(\frac{a}{sin{A}}=\frac{b}{sin{B}}=\frac{c}{sin{C}}=2R\)   kunnen we  c  vervangen door  2R.sin C
en  a  door  2R.sin A.
Dan is \(\large\frac{c-a.cos{B}}{a.sin{B}}=\frac{2Rsin{C}-2Rsin{A}.cos{B}}{2R.sin{A}.sin{B}}=\frac{sin{C}-sin{A}.cos{B}}{sin{A}.sin{B}}=\frac{sin{(}A+B)-sin{A}.cos{B}}{sin{A}.sin{B}}\\ =\frac{sin{A}cos{B}\,+\,cos{A}sin{B}\,-\,sin{A}.cos{B}}{sin{A}.sin{B}}=\frac{cos{A}sin{B}}{sin{A}.sin{B}}=\frac{cos{A}}{sin{A}}=cot{A} \)