SOLUTION: Use the given information to evaluate each expression. (Enter your answers in exact form.) cos &#946; = &#8722;2/7 with 180° < &#946; < 270° (a) sin(&#946;/2) = (b) cos(&#946

Algebra ->  Trigonometry-basics -> SOLUTION: Use the given information to evaluate each expression. (Enter your answers in exact form.) cos &#946; = &#8722;2/7 with 180° < &#946; < 270° (a) sin(&#946;/2) = (b) cos(&#946      Log On


   

Question 975681: Use the given information to evaluate each expression. (Enter your answers in exact form.)
cos β = −2/7 with 180° < β < 270°
(a) sin(β/2) =
(b) cos(β/2) =
(c) tan(β/2) =

Answer by ikleyn(52790) About Me  (Show Source):
You can put this solution on YOUR website!

Use the formulas for trigonometric functions of half argument  (see,  for example,  the lesson  FORMULAS FOR TRIGONOMETRIC FUNCTIONS  in this site)

sin%28beta%2F2%29 = sqrt%28%281-cos%5E2%28beta%29%29%2F2%29 = sqrt%28%281-%28%28-2%29%2F7%29%5E2%29%2F2%29 = sqrt%28%281-%284%2F49%29%29%2F2%29 = sqrt%28%28%2849-4%29%2F49%29%2F2%29 = sqrt%2845%2F%2849%2A2%29%29 = %283%2F7%29%2Asqrt%285%2F2%29 = %283%2F7%29%2A%281%2F2%29%2Asqrt%2810%29 = %283%2F14%29%2Asqrt%2810%29.

cos%28beta%2F2%29 = -sqrt%28%281%2Bcos%5E2%28beta%29%29%2F2%29 = -sqrt%28%281%2B%28%28-2%29%2F7%29%5E2%29%2F2%29 = -sqrt%28%281%2B%284%2F49%29%29%2F2%29 = -sqrt%28%28%2849%2B4%29%2F49%29%2F2%29 = -sqrt%2853%2F%2849%2A2%29%29 = -%281%2F7%29%2Asqrt%2853%2F2%29 = -%281%2F7%29%2A%28sqrt%28106%29%2F2%29 = -%281%2F14%29%2Asqrt%28106%29

                    (the sign  "minus"  because the angle  beta%2F2  is in the 2-nd quadrant).

tan%28beta%2F2%29 = sin%28beta%2F2%29%2Fcos%28beta%2F2%29 = -%28%283%2A14%29%2F14%29%2A%28sqrt%2810%29%2Fsqrt%28106%29%29 = -3%2A%28sqrt%2810%29%2Fsqrt%28106%29%29.