Math Problem Solver

The circle below has radius $1$. Eight segment lengths are labeled with lowercase letters. Six of these equal trigonometric functions of $\theta$. Your answer to this problem should be a six letter sequence whose letters represent the segment lengths that equal the following functions (in the correct order):

\[\sin (\theta), \cos(\theta),\tan(\theta),\csc(\theta),\sec(\theta),\cot (\theta).\][asy]

size(200);

pair O,A,B,C,D;

O = (0,0);

A = rotate(55)*(1,0);

D = intersectionpoint(O--(3,0), A -- (A + scale(2)*rotate(-90)*A));

C = foot(A,O,D);

B = intersectionpoint(D -- (A + A - D), O -- (0,5));

draw(Circle(O,1));

draw(O--A--D--O--B--A--C);

//label("$A$",A,SE);

label("$O$",O,SW);

//label("$B$",B,NW);

//label("$C$",C,NW);

//label("$D$",D,SE);

draw(rightanglemark(D,O,B,2));

draw(rightanglemark(B,A,O,2));

draw(rightanglemark(D,C,A,2));

draw("$a$",(O-(0,0.12))--(C-(0,0.12)),blue,Bars(2mm));

draw("$b$",(D-(0,0.12))--(C-(0,0.12)),S,red,Bars(2mm));

draw("$c$",(O-(0,0.31))--(D-(0,0.31)),brown,Bars(2mm));

label("$d$",(A+C)/2,E);

label("$f$",(B/2),W);

pair T = A/10;

draw("$g$",(B+T)--(A+T),NE,blue,Bars(2mm));

draw("$h$",(A+T)--(D+T),NE,red,Bars(2mm));

draw("$k$",(B+3*T)--(D+3*T),NE,brown,Bars(2mm));

label("$\theta$",O+(0.1,0),NE);

label("$1$",A/2,NW);

[/asy]