I'm wondering if I'm just missing something. I'm following the solution to a related rates problem, and everything makes sense up until one point (screenshot to be attached below).

Question

mendicant_bias · Answer

attachment

mendicant_bias · Answer

I'm assuming I'm just looking over something really small, but I can't see how he could just go from tan(theta) to the value of cos(theta) without either using a trig identity, which wouldn't work here, or just using the pythagorean theorem to find the hypotenuse first.

mendicant_bias · Answer

(Just to be absolutely clear, virtually everything but what's in the red box makes sense to me. I don't need any help understanding the related rates problem itself.)

psymon · Answer

Well, initially cos^2(θ ) came about because after taking the derivative of tan(θ ) to get sec^2(θ ). Then dividing both sides by sec^2(θ ), he changed 1/(sec^2(θ )) into cos^2(θ ). As for the value of cos^2(θ ), that is a pythagorean theorem result. Given tanθ = 1000/500, you can solve for the hypotenuse using those two values. You get a hypotenuse of 500√(5). Then you find cos to be 500/(500√(5)) which gets you 1/√(5). Square it and you get your magical cos^2 number you had.

mendicant_bias · Answer

I get how cos^2(theta) came about. I understand how he got cos^2(theta). Again, I only didn't get how he (quickly) managed to square and factor 1000 and 500. This is a five minute problem, which is why I ask. So, nevermind, he just did use the pythagorean theorem, no FM involved.

phi · Answer

you start with
$$ \tan \theta = \frac{40 t^2}{500} $$
evaluate at t= 5:
$$ \tan \theta = \frac{40 \cdot 5 \cdot 5}{500} = \frac{200 \cdot \cancel{5}}{\cancel{500}100} = \frac{200}{100}=2$$

now draw a right triangle, so that tan θ = 2:
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