In this video I go over another example on sketching polar coordinates and this time sketch the curve r = 1 + sinθ. Unlike in my previous example, instead of plotting points, I instead first sketch the curve on a Cartesian coordinate system and then from there transcribe the curve onto a polar coordinate system. Because the function r = 1 + sinθ is simply sinθ but shifted upwards by 1 on a Cartesian plane, it is easy to determine the points and how the shape would look like on a polar plane. I show in this video step by step how to go about graphing the function on polar coordinates through 4 intervals from θ = 0 to θ = 2pi. Outside of this range, the same path just gets traced over again because of the cyclical nature of the trigonometric function. The shape we end up having is in the shape of a heart, and is known as a Cardioid, which is in fact Greek for the word “heart”. This is a very useful video in showing the steps to follow to systematically graph a circular or trigonometric polar function, so make sure to watch this video!
Watch video on:
- 3Speak:
- Odysee: https://odysee.com/@mes:8/polar-coordinates-example-7-cardioid:7
- BitChute:
- Rumble:
- DTube:
- YouTube:
Download video notes: https://1drv.ms/b/s!As32ynv0LoaIhvBqiGxa0hYLmBxy9g
View Video Notes Below!
Download these notes: Link is in video description.
View these notes as an article: @mes
Subscribe via email: http://mes.fm/subscribe
Donate! :) https://mes.fm/donate
Buy MES merchandise! https://mes.fm/store
More links: https://linktr.ee/matheasy
Follow my research in real-time on my MES Links Telegram: https://t.me/meslinks
Subscribe to MES Truth: https://mes.fm/truthReuse of my videos:
- Feel free to make use of / re-upload / monetize my videos as long as you provide a link to the original video.
Fight back against censorship:
- Bookmark sites/channels/accounts and check periodically
- Remember to always archive website pages in case they get deleted/changed.
Recommended Books:
- "Where Did the Towers Go?" by Dr. Judy Wood: https://mes.fm/judywoodbook
Join my forums!
- Hive community: created/hive-128780
- Reddit: https://reddit.com/r/AMAZINGMathStuff
- Discord: https://mes.fm/chatroom
Follow along my epic video series:
- #MESScience: https://mes.fm/science-playlist
- #MESExperiments: @mes/list
- #AntiGravity: @mes/series
-- See Part 6 for my Self Appointed PhD and #MESDuality breakthrough concept!- #FreeEnergy: https://mes.fm/freeenergy-playlist
- #PG (YouTube-deleted series): @mes/videos
NOTE #1: If you don't have time to watch this whole video:
- Skip to the end for Summary and Conclusions (if available)
- Play this video at a faster speed.
-- TOP SECRET LIFE HACK: Your brain gets used to faster speed!
-- MES tutorial: @mes/play-videos-at-faster-or-slower-speeds-on-any-website- Download and read video notes.
- Read notes on the Hive blockchain #Hive
- Watch the video in parts.
-- Timestamps of all parts are in the description.Browser extension recommendations:
- Increase video speed: https://mes.fm/videospeed-extension
- Increase video audio: https://mes.fm/volume-extension
- Text to speech: https://mes.fm/speech-extension
--Android app: https://mes.fm/speech-android
Polar Coordinates: Example 7: Cardioid
Example 7
Sketch the curve r = 1 + sinθ
Solution
Instead of plotting points as in Example 6, we can first sketch the graph of r = 1 + sinθ in Cartesian coordinates:
This enables us to read at a glance the values of r that correspond to increasing values of θ.
For instance, we see that as θ increases from 0 to π/2, r (the distance from O) increases from 1 to 2, so we sketch the corresponding part of the polar curve in the figure below:
As θ increases from π/2 to π, r decreases from 2 to 1, so we sketch the next part of the curve as in the figure below:
As θ increases from π to 3π/2, r decreases from 1 to 0:
Finally, as θ increases from 3π/2 to 2π, r increases from 0 to 1:
If we let θ increase beyond 2π or decrease beyond 0, we would simply retrace our path because of the periodic nature of the trigonometric sine function.
Putting together all the parts of the curve from above, we get the following shape:
This shape is called a cardioid because it's shaped like a heart!