Step 1 Ans: Given, sin theta = (15/17) ( pi/2) < theta < pi We have to find exact value of cos ( theta/2) Since, sin theta = (Perpendicular/Hypotenuse) Rightarrow Perpendicular = 15 Hypotenuse = 17 We know that, Pythagorean theorem Hypotenuse^(2) = Perpendicular^(2) + Base^(2) Rightarrow (17)^(2) = (15)^(2) + Base^(2) Rightarrow Base^(2) = 289 - 225 = 64 Rightarrow Base = 8 Since, cos theta = (Base/Hypotenuse) Rightarrow cos theta = (8/17) because ( pi)[2) < theta < pi (2^nd quadrant) So, cos theta = (-8/17) Now, We know that cos^(2) theta = 2 cos^(2) theta - 1 Rightarrow cos^(2) theta = (1 + cos^(2) theta/2) Rightarrow cos^(2) ( theta/2) = (1 + cos theta/2) Rightarrow cos ( theta/2) = sqrt( (1 + cos theta/2)) because ( pi/2) < theta < pi Rightarrow ( pi)[4) < ( theta/2) < ( pi/2) (1^(st) quadrant) Rightarrow cos ( theta/2) = sqrt( (1 + ( (-8/17))(2)) = sqrt( (17 - 8/17 *2)) = sqrt( (9/34)) = (3/ sqrt(34)) = (3/ sqrt(34)) * ( sqrt(34)/ sqrt(34)) = (3 sqrt(34)/34) Hense, cos ( theta/2) = (3 sqrt(34)/(34)) rightarrow Barselona, Spain