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I've been playing around with finding the domain-restricted inverses of trigonometric equations using the inverse trigonometric equations. One of the easier formulas I came up with was the formula for the inverse $$a\cos^2x+b\sin^2x$$ The process I used to invert this was to use the pythagorean identities to turn it into a single trigonometric function: $$=a(\cos^2x+\sin^2x)+(b-a)\sin^2x$$ $$=a(1)+(b-a)\sin^2x$$ $$=a+(b-a)\sin^2x$$ and so now I can easily say that the partial inverse is $$\arcsin\sqrt{\frac{x-a}{b-a}}$$ The other form that I managed to find an inversion formula for was the form $$a\cos x+b\sin x$$ My strategy for this one was to use the sum-angle identity after manipulating the expression a bit: $$=\sqrt{a^2+b^2}\bigg(\frac{a}{\sqrt{a^2+b^2}}\cos x+\frac{b}{\sqrt{a^2+b^2}}\sin x\bigg)$$ $$=\sqrt{a^2+b^2}\bigg(\cos x\sin\arcsin\frac{a}{\sqrt{a^2+b^2}}+\sin x\cos\arccos\frac{b}{\sqrt{a^2+b^2}}\bigg)$$ $$=\sqrt{a^2+b^2}\bigg(\cos x\sin\arcsin\frac{a}{\sqrt{a^2+b^2}}+\sin x\cos\arcsin\frac{a}{\sqrt{a^2+b^2}}\bigg)$$ and now I use the sum-angle formula to reduce this to $$=\sqrt{a^2+b^2}\sin\bigg(x+\arcsin\frac{a}{\sqrt{a^2+b^2}}\bigg)$$ and now we can easily find that the inverse is $$\arcsin\bigg(\frac{x}{\sqrt{a^2+b^2}}\bigg)-\arcsin\bigg(\frac{a}{\sqrt{a^2+b^2}}\bigg)$$ However, the last one I've been working with is giving me a little bit of trouble. I can't figure out how to invert $$a\cos x+b\sin^2 x$$ and based on the shapes of its graphs, I suspect some forms can't even be partially inverted this way.

Am I on a wild goose chase? If not, does anybody have any hints?

One more question - does anybody know of other interesting expressions like my two examples that can be inverted? I really enjoy the puzzle of manipulating these expressions to get an inverse, but I don't want to waste my time on any impossible ones.

Thanks!

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  • $\begingroup$ In real arcsin t the must be -1<= t <= 1 a/Sqrt[a^2+b^2] may satisfy or may not depending on a and b $\endgroup$
    – Raffaele
    Commented Jun 29, 2017 at 15:34
  • $\begingroup$ @Raffaele Yes, I know, it doesn't cover all of the same domain that the normal $\arcsin$ does. $\endgroup$ Commented Jun 29, 2017 at 15:39
  • $\begingroup$ Change $\sin^2$ to be in terms of $\cos^2$. Now you have a polynomial in terms of $\cos x$ so apply the quadratic formula. I expect there's a load of faffing about to decide which root to take to get $\cos x$ but you can then apply $\arccos$ $\endgroup$ Commented Jul 9, 2017 at 1:07
  • $\begingroup$ Graphing $a\cos^2x+b\sin^2x$, $~a\cos x+b\sin x$, and $a\cos x+b\sin^2x$ for some values of $a$ and $b$ shows why this is so much harder than the other two $\endgroup$ Commented Jul 12, 2017 at 21:21

1 Answer 1

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Invert the equation $y=a \cos x +b \sin^2 x$ (for $x$)?

Substitute $\sin^2 x = 1- \cos^2 x$, multiply by $-4b$ and complete the square. \begin{eqnarray*} \underbrace{4 b^2 \cos^2 x -4 ab \cos x + a^2}_{(2b \cos x-a)^2} =a^2 +4b^2 -4by \\ x = \cos^{-1} \left( \frac{a \pm \sqrt{a^2+4 b^2 -4 b y}}{2b} \right) \end{eqnarray*} So in keeping with the question (where $x$ and $y$ are swapped ) the formula that you seek is \begin{eqnarray*} \color{red}{\cos^{-1} \left( \frac{a \pm \sqrt{a^2+4 b^2 -4 b x}}{2b} \right)}. \end{eqnarray*}

FURTHER EXAMPLES : Some other nice examples of similar functions that can be inverted are ... \begin{eqnarray*} a \cos x +b \sin 2 x \,\,\,\,\, & \longleftrightarrow & \,\,\,\,\, ? \end{eqnarray*} \begin{eqnarray*} a \tan^2 x +b \sec^2 x \,\,\,\,\, & \longleftrightarrow & \,\,\,\,\, ? \end{eqnarray*} \begin{eqnarray*} a \tan^2 x +b \sec x \,\,\,\,\,\, & \longleftrightarrow & \,\,\,\,\,\, ? \end{eqnarray*} \begin{eqnarray*} a \tan x +b \sec x \,\,\,\,\,\, & \longleftrightarrow & \,\,\,\,\,\, ? \tiny{\text{HINT : convert to $ \cos$ and $\sin$ and then use the $cos$ half} angle formula.} \end{eqnarray*} \begin{eqnarray*} a \cos(hx+j) +b \cos (hx+k) \,\,\,\,\,\, & \longleftrightarrow & \,\,\,\,\,\, ? \end{eqnarray*} And here are the two from the question: \begin{eqnarray*} a \cos x +b \sin x \,\,\,\,\,\, & \longleftrightarrow & \,\,\,\,\,\, ? \end{eqnarray*} \begin{eqnarray*} a \cos^2 x +b \sin^2 x \,\,\,\,\,\, & \longleftrightarrow & \,\,\,\,\,\, ? \end{eqnarray*} And another . \begin{eqnarray*} a \cos x +b \sec x \,\,\,\,\,\, & \longleftrightarrow & \,\,\,\,\,\, ? \end{eqnarray*}

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  • $\begingroup$ I will not up vote you because I asked for a hint, not a full answer. However, I will up vote (and possibly award the bounty) if you can suggest some other similar inversion problems. $\endgroup$ Commented Jul 11, 2017 at 21:48
  • $\begingroup$ He has done that now $\endgroup$ Commented Jul 12, 2017 at 21:45
  • $\begingroup$ @DonaldSplutterwit Thank you, those are very good examples! $\endgroup$ Commented Jul 12, 2017 at 22:19
  • $\begingroup$ @Nilknarf You could replace $\tan$ by $\cot$ and $\sec$ by $\operatorname{cosec}$ (but these would be almost the same.) I am still trying to dream up some more ... let me know find any more ... $\ddot \smile$ $\endgroup$ Commented Jul 12, 2017 at 22:52
  • $\begingroup$ @DonaldSplutterwit Okay! And thanks for the problems, I really appreciate it... I've had about five previous answers to this question that did the same thing you did, but when I refused to up vote and asked for problems, they just deleted their answers and didn't some back. Definitely not deserving of $+150$... I'll probably give it to you, though. :D $\endgroup$ Commented Jul 12, 2017 at 22:54

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