Find the exact value of the expression. Show the rotation and the point on the Unit Circle

Question

cot(^⁄7⁄6 pi) = , csc(⁷⁄4 pi) = , tan(¹⁄6 pi) =

Answer 1

cot(7pi/6) = cot210 = sqr3 = about 1.732

reference angle = 30 or pi/6

csc(7pi/4) = csc(315) = csc(-45) = 1/sin(-45) = -2/sqr2 = -sqr2 = about - 1.414

reference angle = 45 or pi/4

tan(pi/6) = tan(30) = 1/sqr3 = sqr3/3 = about 1.732/3 = 0.577

reference angle = 30 or pi/6

points on a unit circle are

for cot(7pi/6) is (-1/2,-sqr3/2)

for csc(7pi/4) it's (-sqr2/2, -sqr2/2)

for tan(pi/6) it's (sqr3/2, 1/2)

Answer 2

(blue in sketch) cot(7π/6) = adj/opp = (-√3/2)/(-1/2) simplify by dividing these fractions

(red in sketch) csc(7π/4) = hyp/opp = 1/(-√2/2) simplify by dividing fractions and rationalizing the denominator

(green in sketch) tan(π/6) = opp/adj = (1/2)/(√3/2) simplify by dividing fractions and rationalizing the denominator