SOLUTION: Find sin x/2, cos x/2, and tan x/2from the given information. sec x = 9/8, 270° < x < 360°

Algebra ->  Trigonometry-basics -> SOLUTION: Find sin x/2, cos x/2, and tan x/2from the given information. sec x = 9/8, 270° < x < 360°      Log On


   

Question 1131784: Find sin x/2, cos x/2, and tan x/2from the given information. sec x = 9/8, 270° < x < 360°
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
Find sin+%28x%2F2%29, cos%28+x%2F2%29, and+tan+%28x%2F2%29 from the given information:

sec+%28x+%29=+9%2F8, 270° < x < 360°
sec+%28x+%29=+9%2F8=hypotenuse%2Fadjacent_side
hypotenuse=9
adjacent_+side=8
opposite_side=sqrt%289%5E2-8%5E2%29=sqrt%2881-64%29=sqrt%2817%29

sin+%28x%29=-sqrt%2817%29%2F9 (in quadrant IV where sin%28x%29+%3C+0)
cos%28+x%29=8%2F9

Identities you need to use:
sin+%28x%2F2%29= ± sqrt%28%281-cos+%28x%29%29%2F2%29

choose positive root because %28x%2F2%29 is in quadrant II where+sin%28x%29+%3E+0
sin+%28x%2F2%29= ± sqrt%28%281-cos+x%29%2F2%29=sqrt%28%281-8%2F9%29%2F2%29=sqrt%281%2F18%29

cos+%28x%2F2%29= ± sqrt%28%281%2Bcos%28+x%29%29%2F2%29

choose negative root because %28x%2F2%29 is in quadrant II where cos%28x%29%3C+0



How to check answers with calculator:
sec+%28x%29=9%2F8
cos+%28x%29=8%2F9
cos%5E-1%288%2F9%29≈ 0.47588225 radians ≈27.27° (reference angle in specified quadrant IV)

standard position of angle=360-27.27%B0=332.73°
x%2F2=332.73%2F2=166.37º

reference angle=180-166.37=13.63º

sin+%28x%2F2%29=sin+%2813.63%29 ≈0.2357.......(in quadrant II where sin%28x%29+%3E+0)
and above we have
sin+%28x%2F2%29=sqrt%281%2F18%29=0.2357.....which confirms our solution

you can check cos+%28x%2F2%29 and tan %28x%2F2%29 in the same way