MathDB

Given a positive integer

n > 1

and an angle

\alpha < 90^o

, Jaime draws a spiral

OP_0P_1...P_n

of the following form (the figure shows the first steps):

\bullet

First draw a triangle

OP_0P_1

with

OP_0 = 1

\angle P_1OP_0 = \alpha

and

P_1P_0O = 90^o

\bullet

then for every integer

1 \le i \le n

draw the point

P_{i+1}

so that

\angle P_{i+1}OP_i = \alpha

\angle P_{i+1}P_iO = 90^o

and

P_{i-1}

and

P_{i+1}

are in different half-planes with respect to the line

OP_i

https://cdn.artofproblemsolving.com/attachments/f/2/aa3913989dac1cf04f2b42b5d630b2e096dcb6.png a) If

n = 6

and

\alpha = 30^o

, find the length of

P_0P_n

. b) If

n = 2018

and

\alpha= 45^o

, find the length of

P_0P_n

4