The Area of a rectangle can be given in terms of the length p of a diagonal and two trigonometric functions of an angle formed by the diagnoal and a side.Which of the following formulas gives the area A of this rectangle and why?
A. p^2 sec(O)
B. p^2 cot(O)
C. p^2 sin(O) tan(O)
D. p^2 sin(O) cos(O)
all we are doing here, is rearranging equations to get what we want. they asked about area, so we come up with the general formula for the area. since we don't have numbers, we have to use formulas to represent the components that we need.
now, for a rectangle:
A = length * width
now we need to find formulas for the length and width. if you label the vertices of the triangle as i did below, you'd realize that the length is AB and the width is AC
so the area will be given by A = AB*AC
now sine = opp/hyp and cosine = adj/hyp ........the hyp is p
=> sin(o) = AB/p => AB = psin(o)
=> cos(o) = AC/p => AC = pcos(o)
Now, A = AB*AC = psin(o) * pcos(o) = p^2 * sin(o)cos(0)