As shown in the accompanying figure, suppose that lines L1 and L2 form an angle

Question

As shown in the accompanying figure, suppose that lines L1 and L2 form an angle theta, 0< theta < pi/2, at their point of intersection P. A point P0 is chosen that is one L1 and a units from . Starting from P0 a zig-zag path is constructed by successively goinf back and forth between L1 and L2 along a perpendicular from one line to the other. Find the following sums in terms of theta and a.

I know what the answer is but I want to know how to get the answers.

As shown in the accompanying figure, suppose that lines L_1 and L_2 from an angle theta, 0 less than theta less than pie/2, at their point of intersection P. A point p_0 a zig-zag path is constructed by successively going back and forth between L_1 and L_2 along a perpendicular from one line to the other. Find the following sums in terms of theta and a. P_0P_1 + P_1 P_2 + P_2 P_3 + ... P_0 P_1 + P_2 P_3 + P_4 P_5 + ... P_1 P_2 + P_3 P_4 +P_5 P_6 + ...

Explanation / Answer

here angle between two line is pi/2

and from question it is given that p0p1 is perpendicular to line L2 and p1p2 is perpendicular to line L1 and so on.

angle(p0p1p)=pi/2

angle(p1p2p)=pi/2

so,

angle(p1p2p0) is also pi/2

so,

in triangle p0p1p

angle(pp0p1)=(pi/2)-(theta)

so,now using trigonometric identities ,

sin(theta)=p0p1/a

p0p1=a sin(theta)

now in triangle p0p1p2

sin(90-theta)=p1p2/p0p1

cos(theta)=p1p2/p0p1

p1p2=p0p1cos(theta)

simmilarly you can calculate all the required value.

and then add them to get the required pattern.

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