(a) given that theta is an acute angle with sin theta = 17/41 find the exact value of cot theta (b) using the exact values for sine and cosine of both 3*pi/4 and pi/6 and the angle sum identity for sine, find the exact value of sin (11*pi/12)?

1 Answer
Jim G.
Oct 12, 2017

#4/17sqrt87" and "1/4(sqrt6-sqrt2)#

Explanation:

#"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)#

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