Discover the Solution to Sin(arctan(x/4)) with Expert Guidance

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To solve Sin(arctan(x/4)), a right triangle is constructed with sides x and 4, where the angle θ corresponds to arctan(x/4). The tangent of angle θ is defined as x/4, leading to the identification of sin(θ) using the Pythagorean theorem to find the hypotenuse. After calculations, the expression for sin(θ) is derived as x√(x^2 + 16)/(x^2 + 16). This approach effectively illustrates the relationship between the angle and the sine function in the context of trigonometric identities. The solution is confirmed as correct.
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Sin(arctan(x/4))= ?

Been over 2 years since I've done some math, a little help please?
 
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Set up a right triangle with sides x and 4, so that the tangent of one of the angles is x/4, i.e., tanθ=x/4. Then θ=tan^{-1}(x/4) . From the drawing, figure out the value of sinθ.
 
It might help you if you draw a triangle and split the expression above into component parts.

First, how would you triangle look if you were to show what arctan(x/4) meant?
 
Ok I:

Drew a triangle in quadrant 1 to represent x/4 and labeled the angle A for random sake
then used Pythagorean theorem to find the hyp
after solving for sign and rationalizing I came up with:

SinA= (x(√(x^2)+16)/((x^2)+16) ------ √ ending after the first 16
sound right?
 
Yes, ##\sin(A)=x\frac{\sqrt{x^2+16}}{x^2+16}##.

ehild
 
I tried to combine those 2 formulas but it didn't work. I tried using another case where there are 2 red balls and 2 blue balls only so when combining the formula I got ##\frac{(4-1)!}{2!2!}=\frac{3}{2}## which does not make sense. Is there any formula to calculate cyclic permutation of identical objects or I have to do it by listing all the possibilities? Thanks
Since ##px^9+q## is the factor, then ##x^9=\frac{-q}{p}## will be one of the roots. Let ##f(x)=27x^{18}+bx^9+70##, then: $$27\left(\frac{-q}{p}\right)^2+b\left(\frac{-q}{p}\right)+70=0$$ $$b=27 \frac{q}{p}+70 \frac{p}{q}$$ $$b=\frac{27q^2+70p^2}{pq}$$ From this expression, it looks like there is no greatest value of ##b## because increasing the value of ##p## and ##q## will also increase the value of ##b##. How to find the greatest value of ##b##? Thanks
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