SOLUTION: {{{ Let f (x) = 1/x^2 (sqrt(25 + x^2)) }}} {{{ and g(t) = 5 tan(t) for 0 < t < pi/2 }}} Find (f º g)(t) and simplify so that there are no radicals in the function. Write your an

Algebra ->  Trigonometry-basics -> SOLUTION: {{{ Let f (x) = 1/x^2 (sqrt(25 + x^2)) }}} {{{ and g(t) = 5 tan(t) for 0 < t < pi/2 }}} Find (f º g)(t) and simplify so that there are no radicals in the function. Write your an      Log On


   

Question 923726: +Let+f+%28x%29+=+1%2Fx%5E2+%28sqrt%2825+%2B+x%5E2%29%29+ +and+g%28t%29+=+5+tan%28t%29+for+0+%3C+t+%3C+pi%2F2+
Find (f º g)(t) and simplify so that there are no radicals in the function. Write your answer in terms of sin(t) and cos(t).
(f º g)(t) =
There shouldn't be a parentheses around 25+x^2 just couldn't make it without one it seems
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+Let+f+%28x%29+=+4x%2F%28sqrt%281-x%5E2%29%29+ +and+g%28t%29+=+sin%28t%29+for+0+%3C+t+%3C+pi%2F2+
Find (f compose g)(t) and simplify. Write your answer in terms of sin t and cos t.
(f º g)(t) =
Please explain these
Thanks!
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I'll do the first one to get you started


(f º g)(t) is the same as f%28g%28t%29%29. They are just two ways of saying the same exact thing.


f%28x%29+=+1%2F%28x%5E2%2Asqrt%2825+%2B+x%5E2%29%29+ Start with f(x)


f%28g%28t%29%29+=+1%2F%28%28g%28t%29%29%5E2%2Asqrt%2825+%2B+%28g%28t%29%29%5E2%29%29+ Replace every x with g(t)


On the right side, replace all copies of g(t) with 5tan(t)


f%28g%28t%29%29+=+1%2F%2825%2Atan%5E2%28t%29%2Asqrt%2825+%2B+25%2Atan%5E2%28t%29%29%29+ Square 5%2Atan%28t%29 to get 25%2Atan%5E2%28t%29


f%28g%28t%29%29+=+1%2F%2825%2Atan%5E2%28t%29%2Asqrt%2825%281+%2B+tan%5E2%28t%29%29%29%29+ Factor


Break up the root


f%28g%28t%29%29+=+1%2F%2825%2Atan%5E2%28t%29%2A5%2Asqrt%281+%2B+tan%5E2%28t%29%29%29+ Take the square root of 25


f%28g%28t%29%29+=+1%2F%2825%2Atan%5E2%28t%29%2A5%2Asqrt%28sec%5E2%28t%29%29%29+ Use the trig identity 1%2Btan%5E2%28t%29+=+sec%5E2%28t%29


f%28g%28t%29%29+=+1%2F%2825%2Atan%5E2%28t%29%2A5%2Asec%28t%29%29+ Take the square root of sec%5E2%28t%29. Since 0+%3C+t+%3C+pi%2F2, this means sec%28t%29 is positive.


Rewrite tan & sec in terms of sine & cosine.


f%28g%28t%29%29+=+1%2F%28%28125%2Asin%5E2%28t%29%29%2F%28cos%5E3%28t%29%29%29+ Multiply and rearrange terms.


f%28g%28t%29%29+=+%28cos%5E3%28t%29%29%2F%28125%2Asin%5E2%28t%29%29+ Flip the fraction (1/[a/b] = b/a)


Side Note: this only works when sec%28t%29+%3E+0 (in quadrant I and IV)

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Let me know if that helps or not. Thanks.

If you need more help, feel free to email me at jim_thompson5910@hotmail.com

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