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--- mathematics:trigonometric_derivatives [2018/11/25 01:32] – external edit 127.0.0.1
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-**Trigonometric Derivatives**
-This post includes my four page introduction and proof for the derivatives of the sine and cosine functions, and it also includes notes on how those are then used to derive the formulas for the other four derivatives.  Importantly, this contains my personal notes and proofs on the limit as x->0 of [(sin x) / x ] = 1 and the limit as x->0 of [(cos x - 1) / x] = 0, which are both needed, along with the angle sum identity, to determine the derivative of sine.  I remember doing this back in college, however, I base my adult understanding of this proof on Professor Elvis Zap's speedy proof below.  Professor Zap skips tons of steps and proofs within proofs, however, he is spot on and completely avoids using L'Hopital's Rule, which many Professors utilize without ever having rigorously proven.  Additionally, the Sandwich Theorem is intuitive for first year Calculus students and L'Hopital's Rule is not.
-Here are my original lecture notes on the limit of the difference quotient for sine, or in other words, the derivative of sine. See the bottom of the page, and use the angle sum identity.
-{{ :mathematics:tder1.jpg?600 |1}}
-This next page, at the top, is where I evaluate limit as x->0 of [(cos x - 1) / x] = 0, which came right out the evaluation of the limit of the difference quotient of sine.  Below that, you can see my original sketches pertaining to the evaluation of limit as x->0 of [(sin x) / x ] = 1.
-{{ :mathematics:tder2.jpg?600 |2}}
-Having thus found the formula for the derivative of sine, I then used the same method to find the derivative of cosine.  At the bottom of this page below, I then utilized the derivatives for sine and cosine to determine the derivative of tangent using the quotient rule.
-{{ :mathematics:tder3.jpg?600 |3}}
-Similar to how I determined tangent's derivative, I used the quotient rule to determine secant, cosecant, and cotangent's derivatives below.
-{{ :mathematics:tder4.jpg?600 |4}}
-The hardest part of all of the above is obviously the proof of the limit as x->0 of [(sin x) / x ] = 1.  My original notes were used as a jumping off point for me to prove this at the board and were deliberately sparse and personal.  At the same time, when I revisited this proof later, I realized there were a great deal many steps that I glossed over, missed entirely, or simply did not take the time to prove or remind the students of how those results were obtained.  For that reason, I have included the expanded and full proof of the limit as x->0 of [(sin x) / x ] = 1 below.  This was written in the summer of 2015.
-The first part of the expanded proof covers the basic geometry of the construction using the unit circle, the relationship and determination of the areas of each of the regions under consideration, and it also includes formal geometric language and secondary proofs to obtain those results.  It is important to note that theta > 0 for the first two pages of the proof.
-{{ :mathematics:tder5.jpg?600 |5}}
-The second part uses the three areas that have now been calculated and sets up a compound inequality, splits it apart and states each branch of the inequality in relation to [(sin x) / x ]  and then after re-combining them into a new and equivalent inequality, slaps the limit operator on those expressions and evaluates the compound inequality using the Sandwich Theorem.
-{{ :mathematics:tder6.jpg?600 |6}}
-The last page goes into detail about why the inverse limit theorem works for odd functions only, whereby you switch the sign on limit's element and switch the sign on the limit's expression, and always end up with the same numerical result for the evaluated limit.  Again, this only works for odd functions.  I gave two additional examples as an intuitive demonstration.  Lastly, I used the definition of an odd function to pull the negative out of the angle of sine and apply it to the amplitude, then after simplifying, used the transitive property to demonstrate that when theta < 0, the limit results in the same evaluation.
-{{ :mathematics:tder7.jpg?600 |7}}
-Lastly, here is Professor Elvis Zap's original YouTube video and super speedy proof of the limit as x->0 of [(sin x) / x ] = 1.  He is pretty awesome.  You should like his video and subscribe to his channel.
-[[https://www.youtube.com/embed/o6S6RbfhRTU|Professor Zap's Proof]]
- --- //[[netcmnd@jonathanhaack.com|oemb1905]] 2017/05/16 02:09//

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