https://math.stackexchange.com/questions/4985471/visual-proof-of-harmonic-form-results Skip to main content Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Visit Stack Exchange [ ] Loading... 1. + Tour Start here for a quick overview of the site + Help Center Detailed answers to any questions you might have + Meta Discuss the workings and policies of this site + About Us Learn more about Stack Overflow the company, and our products 2. 3. current community + Mathematics help chat + Mathematics Meta your communities Sign up or log in to customize your list. more stack exchange communities company blog 4. 5. Log in 6. Sign up Mathematics 1. 1. Home 2. Questions 3. Tags 4. 5. Users 6. Unanswered 2. Teams [teams-promo] Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Try Teams for free Explore Teams 3. Teams 4. Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Explore Teams Teams Q&A for work Connect and share knowledge within a single location that is structured and easy to search. Learn more about Teams Visual Proof of Harmonic Form Results Ask Question Asked 2 months ago Modified 2 months ago Viewed 511 times 1 $\begingroup$ When you have $A\sin(x) + B\cos(x)$, I'm aware that is $\sqrt{A^2 + B ^2}\sin(x+\arctan(b/a))$. I know how to prove this with algebra, but I was wondering if there was a visual proof to show this off. I personally began by considering the case $3 \sin(x) + 4 \cos(x)$ to get my feet on the problem, getting an ellipse and considering "maximizing the sum of the coordinates on the edge of it". However, that became unfruitful quickly. * geometry * trigonometry * visualization Share Cite Follow edited Oct 16 at 19:02 Bml's user avatar Bml 1,77422 gold badges88 silver badges2222 bronze badges asked Oct 16 at 16:54 AaravG's user avatar AaravGAaravG 8355 bronze badges $\endgroup$ 1 * $\begingroup$ Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. $\endgroup$ - Community Bot Commented Oct 16 at 17:09 Add a comment | 3 Answers 3 Sorted by: Reset to default [Highest score (default) ] 5 $\begingroup$ Am I over 30 characters (minimum answer size) yet? enter image description here Share Cite Follow edited Oct 16 at 17:46 answered Oct 16 at 17:41 Soham Saha's user avatar Soham SahaSoham Saha 1,89944 silver badges2626 bronze badges $\endgroup$ Add a comment | 2 $\begingroup$ Here's a non-overlapping variant of @SohamSaha's diagram (and a stylistic variant of a diagram shown in this ancient answer of mine): harmonic trigonograph $$\begin{align} a\sin\theta+b\cos\theta &= c \sin(\theta+\phi) \\ [4pt] a\cos\theta-b\sin\theta &= c \cos(\theta+\phi) \end{align}$$ $$a^2 + b^2 = c^2 \qquad \tan\phi = \frac{b}{a}$$ Share Cite Follow edited Oct 17 at 2:33 answered Oct 16 at 18:24 Blue's user avatar BlueBlue 80.3k1414 gold badges124124 silver badges247247 bronze badges $\endgroup$ Add a comment | 1 $\begingroup$ $4\cos(x)+3\sin(x)$ is the dot product of the fixed vector $(4,3)$ and the rotating unit vector $(\cos(x),\sin(x))$. This dot product is the product of the lengths of the two vectors ($1$ and $5$), times the cosine of the angle they form. It is also the product of the length of the fixed vector by the algebraic length of the projection of the rotating vector. enter image description here If you rotate the figure to bring the fixed vector vertical, you get a purely sinusoidal function. enter image description here Share Cite Follow answered Oct 16 at 19:24 user1353692user1353692 $\endgroup$ Add a comment | You must log in to answer this question. Not the answer you're looking for? Browse other questions tagged * geometry * trigonometry * visualization . * Featured on Meta * The December 2024 Community Asks Sprint has been moved to March 2025 (and... * Stack Overflow Jobs is expanding to more countries Linked 2 Summation of trigonometric functions Related 0 Newbie with barycentric coordinates: why one is zero when on a vertex? 2 How do I find 2x2 orthonormal diagonalizing matrices using only trigonometry? 9 Converting a rotated ellipse in parametric form to cartesian form 0 trigeome-Tree question: using calculus 3 Prove $\lceil z \rceil=z+\frac12-\frac{\tan^{-1}(\tan(\pi(z+0.5)))}{\ pi}$ when $z$ is not an integer 1 If $\cos x=\frac{12}{13}$ and $\sin y=\frac{7}{25}$ then what is $\ cos y$ and $\sin x$? 1 Arc length of circle overlaping with ellipse 4 Clock angle problem from simulation of planetary orbits 5 Trouble with tedious algebra (Oxford 1992 Admissions Test 2 1992) Hot Network Questions * Multi-ring buffers of uneven sizes in QGIS * In D&D, do the gods embodying various alignments *see* themselves as embodying and advocating for those alignments along their other interests? * TikZ clipping on polygons clips everything * Is there a repository of WKT definitions for different non-Earth CRS? * How to fix volume distribution (geo nodes)> * M2 storage, PCIe v.3 vs. v.4, does it matter IO/s wise? * XGBoost - order of columns seen by a tree with `colsamle_bytree `<1 * What are the main views on the question of the relation between logic and human cognition? * Removing Z coordinate from GeoJSON using QGIS * Is there a reason that the McCallister house has a doggie door? * Whose logo for an invited seminar in another university? * Why are languages commonly structured as trees? * Handling a customer that is contacting my subordinates on LinkedIn demanding a refund (already given)? * Field type link limited to 255 characters in List - how to work around this issue? * A mysterious commutativity condition for division ring * Joining two lists by matching elements of the two * AWS EC2 : Ubuntu npm warn tar TAR_BAD_ARCHIVE: Unrecognized archive format * How do I install a small pet door in a hollow interior door? Do I need to purchase a solid door to do this installation? * Quiz interface based on Yaml files * Is it true that only prosecutors can 'cut a deal' with criminals? * What are the maximum bonuses of each type possible? * Marginal effect of poisson regression with interaction * Revise & Resubmit: changing the text color of revisions in the text? * Possible to discretize / triangulate a polygon in RegionProduct - style? more hot questions Question feed Subscribe to RSS Question feed To subscribe to this RSS feed, copy and paste this URL into your RSS reader. [https://math.stackex] * Mathematics * Tour * Help * Chat * Contact * Feedback Company * Stack Overflow * Teams * Advertising * Talent * About * Press * Legal * Privacy Policy * Terms of Service * Cookie Settings * Cookie Policy Stack Exchange Network * Technology * Culture & recreation * Life & arts * Science * Professional * Business * API * Data * Blog * Facebook * Twitter * LinkedIn * Instagram Site design / logo (c) 2024 Stack Exchange Inc; user contributions licensed under CC BY-SA . rev 2024.12.17.20590