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I had a simple thing to compute with a calculator: $$\sin\left(2\cos^{-1}\left(\frac{15}{17}\right)\right)$$ I got the decimal answer of about $0.83044983$, but when I typed it in WolframAlpha, it also gave an exact answer of $\frac{240}{289}$. How in the world would one get an exact answer here?

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    $\begingroup$ Hint: $15^2+8^2=17^2$ $\endgroup$ Commented Mar 19, 2018 at 21:38
  • $\begingroup$ @PrasunBiswas actually $8^2$ instead of $6^2$. I knew that, but still wasn't able to do it, because as it turns out i forgot about double angle formula $\endgroup$
    – KKZiomek
    Commented Mar 19, 2018 at 21:40
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    $\begingroup$ Hint #2: $2\times\dfrac 8{17}\times\dfrac{15}{17}=\dfrac{240}{289}$ $\endgroup$ Commented Mar 19, 2018 at 21:41
  • $\begingroup$ how did this simple question get that many upvotes? $\endgroup$ Commented Aug 4, 2018 at 17:45
  • $\begingroup$ @James I have no idea lol, I was thinking the same thing $\endgroup$
    – KKZiomek
    Commented Aug 8, 2018 at 19:54

7 Answers 7

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Here are three relevant formulas:

  • $\sin 2x = 2 \sin x \cos x$.

  • $\cos \cos^{-1} x = x.$

  • $\sin \cos^{-1} x = \sqrt{1 - x^2}$ after drawing an appropriate right triangle.

Combining these three to get the desired conclusion is left to the interested reader.

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    $\begingroup$ My students don't reliably remember the condition on your second equality, so I'd recommend including it. ", if $-1 \leq x \leq 1$." $\endgroup$ Commented Mar 20, 2018 at 15:13
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    $\begingroup$ @EricTowers I might be one of those stupid students in this case, but if you allow complex numbers, doesn't that second equality hold? Or is there some multivalued function weirdness going on that you have to care about? $\endgroup$ Commented Mar 20, 2018 at 21:00
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    $\begingroup$ Not "stupid" -- focusing on the "big", misremembering the "small". Since this Question is tagged "trigonometry", we're very likely talking about real-valued functions. The domain of real-valued arccosine is $[-1,1]$. (The usual extension to the complex plane has branch cuts along the rest of the real line. There is definitely some multivalued weirdness going on; cosine isn't injective, so this is unavoidable.) In fact, in Trig, any time a student uses the shortcut you write, they are expected to explicitly write that they have verified that $x$ is in the domain of arccosine. $\endgroup$ Commented Mar 21, 2018 at 0:01
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    $\begingroup$ @EricTowers: Arccos is multivalued, but for all possible values $y$ of $\cos^{-1}x$ you have $\cos y=x$, by definition. Thus, in complex numbers $\cos\cos^{-1}x=x$ is always true. The converse is not - you could have $\cos^{-1}\cos x\neq x$ if you chose the wrong branch. Whether students in a trig class should first show that $-1\le x\le1$, is more a question of pedagogy than mathematics. $\endgroup$ Commented Mar 21, 2018 at 10:18
  • $\begingroup$ The same is with other inverse functions - for any branch of $\sqrt{x}$, you always have $\left(\sqrt{x}\right)^2=x$, but $\sqrt{x^2}$ can be either $x$ or $-x$ depending on $x$ and the branch. $\endgroup$ Commented Mar 21, 2018 at 10:22
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Call

$$ u = \cos^{-1}\frac{15}{17} $$

Therefore

$$ \cos u = \frac{15}{17} $$

and

$$ \sin u = \sqrt{1 - \cos^2 u} = \sqrt{1 - \frac{15^2}{17^2}} = \frac{8}{17} $$

With these two you just need to calculate

$$ \sin 2u = 2\sin u \cos u = 2\frac{15}{17}\frac{8}{17} = \color{blue}{\frac{240}{289}} $$

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\begin{align} \sin \theta &= \dfrac{8}{17} \\ \cos \theta &= \dfrac{15}{17} \\ \hline \sin\left(2 \arccos \dfrac{15}{17} \right) &= \sin(2 \theta) \\ &= 2 \sin(\theta) \cos(\theta) \\ &= \cdots \end{align}

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  • $\begingroup$ Nice idea.͏͏͏͏͏͏ $\endgroup$
    – Pacerier
    Commented Mar 21, 2018 at 20:01
  • $\begingroup$ I love it! Draw the triangle! $\endgroup$
    – David Dyer
    Commented Mar 23, 2018 at 19:00
  • $\begingroup$ I didn't notice this answer before, but it deserves more points, I like that idea. If I could give two answers best answer, you would be the best answer too $\endgroup$
    – KKZiomek
    Commented Apr 18, 2018 at 14:27
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$ (8,15,17)$ are lengths of a Pythagorean triple right triangle. A narrow right triangle of these side lengths can be drawn if needed.

$$\sin(2\cos^{-1}\frac{15}{17}) = \sin(2\sin^{-1}\frac{8}{17}) = 2 \cdot \frac{8}{17}\cdot \frac{15}{17} =\frac{240}{289}.$$

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Let $\cos^{-1}x=y\implies\cos y=x$

Using Principal values, $0\le x\le\pi\implies\sin y\ge0$

and $\sin y=+\sqrt{1-\cos^2y}=?$

Finally, $\sin2(\cos^{-1}x)=\sin2y=2\sin y\cos y=?$

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Note that $$\sin\left(2\cos^{-1}(a)\right)=2\sin\left(\cos^{-1}(a)\right)\cos\left(\cos^{-1}(a)\right)=2a\sin\left(\cos^{-1}(a)\right)$$ and use the fact that $$\sin\left(\cos^{-1}\left(\frac{\text{adjacent}}{\text{hypotenuse}}\right)\right)=\frac{\text{opposite}}{\text{hypotenuse}}$$

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Hint.....$$\sin2x=2\sin x\cos x$$

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