MHB Trigonometric Simplification for Projectile Motion Equation

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The discussion revolves around simplifying a trigonometric equation related to projectile motion. The original equation involves terms with respect to the variable f and includes a substitution for L. A participant confirms that the equation can indeed be simplified to a form involving secant and sine functions, using key trigonometric identities. The simplification process requires careful manipulation of the terms, highlighting the importance of recognizing fundamental trigonometric relationships. The equation is part of a broader study on projectile motion under specific gravitational conditions.
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Hi!

Long ago (more than 5 years now, actually) I got stuck with a trigonometrig formula and I haven't been able to got the point. I had an equation (with respect to $$f$$):

$$(r - a)L^2 + r\tan \alpha \cos f \cdot L + a\tan^2 \alpha = 0$$,

where

$$L = \sin f - \tan \alpha \cos f$$.

According to my partially lost notes, some CAS has been simplified this into

$$\tfrac{1}{2} \sec ^2 \alpha \sin f [(2a - r)\sin (2\alpha - f) + r\sin f] = 0$$.

Now I'd like to ask you, is this true at all? And if it is true, what tricks(?) were used? Thank you so much! ^^
 
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Theia said:
Hi!

Long ago (more than 5 years now, actually) I got stuck with a trigonometrig formula and I haven't been able to got the point. I had an equation (with respect to $$f$$):

$$(r - a)L^2 + r\tan \alpha \cos f \cdot L + a\tan^2 \alpha = 0$$,

where

$$L = \sin f - \tan \alpha \cos f$$.

According to my partially lost notes, some CAS has been simplified this into

$$\tfrac{1}{2} \sec ^2 \alpha \sin f [(2a - r)\sin (2\alpha - f) + r\sin f] = 0$$.

Now I'd like to ask you, is this true at all? And if it is true, what tricks(?) were used? Thank you so much! ^^

Hey Theia! ;)

Let's see...
$$(r - a)(\sin f - \tan α \cos f)^2 + r\tan α \cos f \cdot (\sin f - \tan α \cos f) + a\tan^2 α = 0 \\
(r-a)(\sin^2 f - 2\tan α \sin f\cos f + \tan^2α \cos^2 f) + r\tan α \cos f \sin f - r\tan^2 α \cos^2 f + a\tan^2 α = 0 \\
(2a-r)\tan α \cos f \sin f + a(-\sin^2 f - \tan^2α \cos^2 f + \tan^2 α)+ r\sin^2 f = 0 \\
(2a-r)\tan α \cos f \sin f + a(-\sin^2 f + \tan^2 α(1 - \cos^2 f )) + r\sin^2 f = 0 \\
(2a-r)\tan α \cos f \sin f - a (1 - \tan^2 α)\sin^2 f + r\sin^2 f = 0 \\
\frac 12 \sec^2 α \sin f\Big[(2a-r)2\sin α \cos α\cos f - 2a (\cos^2 α - \sin^2 α)\sin f + 2r\cos^2 α\sin f \Big] = 0 \\
\frac 12 \sec^2 α \sin f\Big[(2a-r)(2\sin α \cos α\cos f - (\cos^2 α - \sin^2 α)\sin f) - r (\cos^2 α - \sin^2 α)\sin f + 2r\cos^2 α\sin f \Big] = 0 \\
\frac 12 \sec^2 α \sin f\Big[(2a-r)(\sin 2α \cos f - \cos 2α\sin f) + r (\cos^2 α + \sin^2 α)\sin f \Big] = 0 \\
\frac 12 \sec^2 α \sin f\Big[(2a-r)\sin(2α - f) + r \sin f \Big] = 0 \\
\sin f = 0 \quad\vee\quad (2a-r)\sin(2α - f) + r \sin f = 0 \\
$$
The "tricks" that we've used are:
$$
\cos^2 α + \sin^2 α = 1 \\
\sin 2α = 2\sin α\cos α \\
\cos 2α = \cos^2 α - \sin^2 α \\
\sin(α-β)=\sin α\cos β - \cos α \sin β
$$
 
Wow! Thank you so much! It looks like the key was some not so obvious manipulations to the equation. *wonders why it is so difficult to add 0 sometimes... :D *

As a side note, this equation was part of my study how to describe projectile motion under Newtonian $$-\gamma \tfrac{Mm}{r^2}$$ gravity (e.g. on Moon) without messing too much with celestial mechanics. Everything else went smoothly, except this simplification... But once the numerics worked, I didn't pay attention to it until now. :D Thank you again!
 
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