he66666
  • he66666
Trig Identity question? The angle x lies in the interval pi/2≤x≤pi, and sin²x=4/9. Determine cos(x/2). answer: √[(3-√5)/6] How do you solve this question (using trig identities)? It gets confusing because it's x/2, not just x. I keep getting 7/9 as my answer..
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
\[cos(\frac{x}{2})=\pm \sqrt{\frac{1+cos(x)}{2}}\]
anonymous
  • anonymous
square root should be over whole thing.
he66666
  • he66666
How does it have a square root? :S Is it derived from a trig identity?

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More answers

anonymous
  • anonymous
\[sin^2(x)=\frac{4}{9}\] \[sin(x)=\pm \frac{3}{4}\]
anonymous
  • anonymous
derived from "double angle" formula \[cos(2x)=2cos^2(x)-1\] replace 2x by x t so x gets replaced by \[\frac{x}{2}\] and then solve for \[cos(\frac{x}{2})\]
anonymous
  • anonymous
\[cos(x)=2cos^2(\frac{x}{2})-1\] \[cos(x)+1=2cos^2(\frac{x}{2})\] \[\frac{cos(x)-1}{2}=cos^2(\frac{x}{2})\]
anonymous
  • anonymous
typo should be \[\frac{cos(x)+1}{2}=cos^2(\frac{x}{2})\]
anonymous
  • anonymous
then take the square root of the whole thing. plus or minus of course. i cannot seem to type set it.
anonymous
  • anonymous
in case \[sin(x)=\frac{3}{4}\] do you know how to find \[cos(x)\]?
he66666
  • he66666
yes, pythagorean theorem?
anonymous
  • anonymous
whoa hold the phone i am tired. it is \[sin(x)=\frac{4}{9}\]
anonymous
  • anonymous
damn \[sin^2(x)=\frac{4}{9}\] \[sin(x)=\frac{2}{3}\]
anonymous
  • anonymous
draw a triangle opposite side 2 and hypotenuse 3 my mistake sorry
anonymous
  • anonymous
adjacent side is \[\sqrt{3^2-2^2}=\sqrt{9-4}=\sqrt{5}\]
anonymous
  • anonymous
so \[cos(x)=\frac{\sqrt{5}}{3}\]
anonymous
  • anonymous
now plug in \[\frac{\sqrt{5}}{3}\] in the formula up top to get your answer. i will write it if you like.
he66666
  • he66666
no it's alright, I got it. Thanks so much for the help satellite! :)
anonymous
  • anonymous
oh another mistake i forgot x was in quad II so cosine is negative. it is \[-\frac{\sqrt{5}}{3}\]
anonymous
  • anonymous
make sure you use \[-\frac{\sqrt{5}}{3}\]
he66666
  • he66666
alright, I will :) thanks
anonymous
  • anonymous
welcome

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