find cos theta/2

Question

if tan theta=3/4, pi<theta<3pi/2

Answer 1

pi<theta<3pi/2

implies pi/2<theta/2<3pi/4

for theta/2 therefore cos <0 and sin >0 (second quadrant)

for theta cos(theta)=-4/5 since tan(theta)=3/4 the hypotenuse is 5 and x coordinate -4

sin(theta/2)= +√(1-(-4/5)) the ambiguity in front of square root is resolved to + (second quadrant)

cos(theta/2)= -√(1+(-4/5)) the ambiguity in front of square root is resolved to - (second quadrant)

tan(theta/2)=[sin(theta/2)]/cos(theta/2)]

Answer 2

There are 2 ways to answer this problem.

First you can look up the angle given as 216.86 degrees, divide it by 2 and look up the cos of that angle: -0.316 .

Otherwise you can that the cos theta=-4/5 (Pythagorean theorem)

and use the 1/2 angle formula to the the cos of that angle.