SOLUTION: Given sin(&#945;)=&#8722;8/9 and 3&#960;/2<&#945;<2&#960;, find the exact value of sin(&#945;/2). Note: You can't use decimals in your answer.

Algebra ->  Trigonometry-basics -> SOLUTION: Given sin(&#945;)=&#8722;8/9 and 3&#960;/2<&#945;<2&#960;, find the exact value of sin(&#945;/2). Note: You can't use decimals in your answer.      Log On


   

Question 1102856: Given sin(α)=−8/9 and 3π/2<α<2π, find the exact value of sin(α/2).
Note: You can't use decimals in your answer.
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
sin%28alpha%2F2%29=sqrt%28%281-cos%28alpha%29%29%2F2%29
So,
cos%5E2%28alpha%29%2Bsin%5E2%28alpha%29=1
cos%5E2%28alpha%29%2B%28-8%2F9%29%5E2=1
cos%5E2%28alpha%29%2B64%2F81=1
cos%5E2%28alpha%29=81%2F81-64%2F81
cos%5E2%28alpha%29=17%2F81
cos%28alpha%29=sqrt%2817%29%2F9 <-- Cosine is positive in Q4.
So then,
sin%28alpha%2F2%29=sqrt%28%281-sqrt%2817%29%2F9%29%2F2%29
Since alpha is in Q4, alpha%2F2 will be in Q2 where sine is positive.