#"using the "color(blue)"trigonometric identities"#
#•color(white)(x)cottheta=costheta/sintheta#
#•color(white)(x)sin^2theta+cos^2theta=1#
#rArrcostheta=+-sqrt(1-sin^2theta)#
#(a)#
#"since theta is acute then theta is in the first quadrant"#
#"where all trig ratios are positive"#
#sintheta=17/41#
#rArrcostheta=sqrt(1-(17/41)^2)#
#color(white)(rArrcostheta)=sqrt(1-(289/1681))=sqrt(1392/1681)=(4sqrt87)/41#
#sqrt1392=sqrt(16xx3xx29)larrcolor(blue)"product of prime factors"#
#rArrsqrt1392=4sqrt87#
#rArrcottheta=(4sqrt87)/cancel(41)xxcancel(41)/17=4/17sqrt87#
#(b)#
#"using the "color(blue)"addition formula for sine"#
#•color(white)(x)sin(A+-B)=sinAcosB+-cosAsinB#
#rArrsin((11pi)/12)=sin((3pi)/4+pi/6)#
#=sin((3pi)/4)cos(pi/6)+cos((3pi)/4)sin(pi/6)#
#[sin((3pi)/4)=sin(pi/4)" and "cos((3pi)/4)=-cos(pi/4)]#
#=sin(pi/4)cos(pi/6)-cos(pi/4)sin(pi/6)#
#"using "color(blue)"exact values"#
#•color(white)(x)sin(pi/4)=cos(pi/4)=1/sqrt2#
#•color(white)(x)sin(pi/6)=1/2" and "cos(pi/6)=sqrt3/2#
#=(1/sqrt2xxsqrt3/2)-(1/sqrt2xx1/2)#
#=sqrt3/(2sqrt2)-1/sqrt2#
#=(sqrt3-1)/(2sqrt2)larrxxsqrt2/sqrt2#
#=1/4(sqrt6-sqrt2)#