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March 8th, 2018, 03:55 PM   #1
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Hello, I would like to know if you can help me with this problem the intergal of (cos (6x)+1), I’ve understood it as far as cos (1/2 (ax)) but where is the sqrt of 2 coming from?! Thank you.
Attached Images 92EE7A3D-54BC-4C56-B7A7-AEAE2C5F087E.jpg (9.1 KB, 8 views) March 8th, 2018, 05:03 PM #2 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,616 Thanks: 2606 Math Focus: Mainly analysis and algebra \begin{align*}\cos{(2A)} &= \cos^2{(A)} - \sin^2{(A)} \\ &= \cos^2{(A)} - \big(1 - \cos^2{(A)}\big) \\ &= 2\cos^2{(A)} - 1 \\ \cos{(2A)} + 1 &= 2\cos^2{(A)} \\[8pt] \sqrt{\cos{(2A)} + 1} &= \sqrt{2\cos^2{(A)}} \\ &= \sqrt{2\vphantom{\cos^2{(A)}}}\sqrt{\cos^2{(A)}} \\ &= \sqrt{2}\cos{(A)}\end{align*} Last edited by v8archie; March 8th, 2018 at 05:05 PM. March 8th, 2018, 05:17 PM   #3
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Oh my goodness, thank you so much for your help, I greatly appreciate it. I’ve been trying to figure this out for so long. I have and solution, but is this the same as this square root of 2, I understand where the 1/3 (4 square root of 2 comes from).
Attached Images 4B15F1A9-578C-4389-B04B-DB392D4FD0E8.jpg (9.9 KB, 1 views) March 8th, 2018, 06:53 PM #4 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,616 Thanks: 2606 Math Focus: Mainly analysis and algebra The $\sqrt2$ comes from the derivation above. It gets multiplied by the $1$ in the following. \begin{align*}\sin{(3x)} &= \sin{(2x)}\cos{(x)} + \cos{(2x)}\sin{(x)} \\ &= \big(2\sin{(x)}\cos{(x)}\big)\cos{(x)} + \big(2\cos^2{(x)}-1\big)\sin{(x)} \\ &= 4\sin{(x)}\cos^2{(x)} - \sin{(x)} \\ &= \big(4\cos^2{(x)} - 1\big)\sin{(x)} \end{align*} I don't know why Symbolab decided to write the answer in trigonometric functions of $x$. I'd have left it in terms of trigonometric functions of $3x$. Last edited by v8archie; March 8th, 2018 at 06:55 PM. Tags question Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post sCoRPion Differential Equations 0 March 16th, 2015 01:05 PM pudro Algebra 0 November 10th, 2008 08:12 PM

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