Difference between revisions of "Derivatives of the Trigonometric Functions"

From Department of Mathematics at UTSA
Jump to navigation Jump to search
(Added links for PowerPoint and worksheet)
 
 
(One intermediate revision by one other user not shown)
Line 1: Line 1:
 +
 +
Sine, cosine, tangent, cosecant, secant, cotangent. These are functions that crop up continuously in mathematics and engineering and have a lot of practical applications. They also appear in more advanced mathematics, particularly when dealing with things such as line integrals with complex numbers and alternate representations of space like spherical and cylindrical coordinate systems.
 +
 +
We use the definition of the derivative, i.e.,
 +
:<math>f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}</math> ,
 +
to work these first two out.
 +
 +
Let us find the derivative of sin(''x''), using the above definition.
 +
 +
{| WIDTH="75%"; style="background-color: #FFFFFF; border: solid 1px #D6D6FF; padding: 1em;" valign="top" |
 +
|-
 +
|<math>f(x)=\sin(x)</math>
 +
|-
 +
|<math>f'(x)=\lim_{h\to 0}\frac{\sin(x+h)-\sin(x)}{h}</math>|| Definition of derivative
 +
|-
 +
|<math>=\lim_{h\to 0}\frac{\cos(x)\sin(h)+\cos(h)\sin(x)-\sin(x)}{h}</math>|| trigonometric identity
 +
|-
 +
|<math>=\lim_{h\to 0}\frac{\cos(x)\sin(h)+(\cos(h)-1)\sin(x)}{h}</math>|| factoring
 +
|-
 +
|<math>=\lim_{h\to 0}\frac{\cos(x)\sin(h)}{h}+\lim_{h\to 0}\frac{(\cos(h)-1)\sin(x)}{h}</math>|| separation of terms
 +
|-
 +
|<math>=\cos(x)\times 1+\sin(x)\times 0</math>|| application of limit
 +
|-
 +
|<math>=\cos(x)</math>|| solution
 +
|-
 +
|}
 +
 +
Now for the case of cos(''x'').
 +
{| WIDTH="75%"; style="background-color: #FFFFFF; border: solid 1px #D6D6FF; padding: 1em;" valign="top" |
 +
|-
 +
|<math>f(x)=\cos(x)</math>
 +
|-
 +
|<math>f'(x)=\lim_{h\to 0}\frac{\cos(x+h)-\cos(x)}{h}</math>|| Definition of derivative
 +
|-
 +
|<math>=\lim_{h\to 0}\frac{\cos(x)\cos(h)-\sin(h)\sin(x)-\cos(x)}{h}</math>|| trigonometric identity
 +
|-
 +
|<math>=\lim_{h\to 0}\frac{\cos(x)(\cos(h)-1)-\sin(x)\sin(h)}{h}</math>|| factoring
 +
|-
 +
|<math>=\lim_{h\to 0}\frac{\cos(x)(\cos(h)-1)}{h}-\lim_{h\to 0}\frac{\sin(x)\sin(h)}{h}</math>|| separation of terms
 +
|-
 +
|<math>=\cos(x)\times 0-\sin(x)\times 1</math>|| application of limit
 +
|-
 +
|<math>=-\sin(x)</math>||solution
 +
|}
 +
 +
Therefore we have established
 +
 +
{| WIDTH="75%"
 +
|-
 +
| style="background-color: #FFFFFF; border: solid 1px #D6D6FF; padding: 1em;" valign="top" |
 +
<center>'''Derivative of Sine and Cosine'''<br>
 +
<math>\frac{d}{dx}\sin(x)=\cos(x)</math><br>
 +
<math>\frac{d}{dx}\cos(x)=-\sin(x)</math><br>
 +
</center>
 +
|}
 +
 +
 +
 +
To find the derivative of the tangent, we just remember that:
 +
 +
<math>\tan(x)=\frac{\sin(x)}{\cos(x)}</math>
 +
 +
which is a quotient. Applying the quotient rule, we get:
 +
 +
<math>\frac{d}{dx}\tan(x)=\frac{\cos^2(x)+\sin^2(x)}{\cos^2(x)}</math>
 +
 +
Then, remembering that <math>\cos^2(x)+\sin^2(x)=1</math> , we simplify:
 +
{|
 +
|-
 +
|<math>\frac{\cos^2(x)+\sin^2(x)}{\cos^2(x)}</math>
 +
|<math>=\frac{1}{\cos^2(x)}</math>
 +
|-
 +
|
 +
|<math>=\sec^2(x)</math>
 +
|-
 +
|}
 +
 +
 +
{| WIDTH="75%"
 +
|-
 +
| style="background-color: #FFFFFF; border: solid 1px #D6D6FF; padding: 1em;" valign="top" |
 +
<center>'''Derivative of the Tangent'''<br>
 +
<math>\frac{d}{dx}\tan(x)=\sec^2(x)</math><br>
 +
</center>
 +
|}
 +
 +
For secants, we again apply the quotient rule.
 +
 +
:<math>\sec(x)=\frac{1}{\cos(x)}</math>
 +
:<math>\begin{align}\frac{d}{dx}\sec(x)&=\frac{d}{dx}\frac{1}{\cos(x)}\\
 +
&=\frac{\cos(x)\frac{d 1}{dx}-1\frac{d\cos(x)}{dx}}{\cos(x)^2}\\
 +
&=\frac{\cos(x)0-1(-\sin(x))}{\cos(x)^2}
 +
\end{align}</math>
 +
 +
Leaving us with:
 +
 +
:<math>\frac{d}{dx}\sec(x)=\frac{\sin(x)}{\cos^2(x)}</math>
 +
 +
Simplifying, we get:
 +
 +
 +
{| WIDTH="75%"
 +
|-
 +
| style="background-color: #FFFFFF; border: solid 1px #D6D6FF; padding: 1em;" valign="top" |
 +
<center>'''Derivative of the Secant'''<br>
 +
<math>\frac{d}{dx}\sec(x)=\sec(x)\tan(x)</math><br>
 +
</center>
 +
|}
 +
 +
Using the same procedure on cosecants:
 +
:<math>\csc(x)=\frac{1}{\sin(x)}</math>
 +
 +
We get:
 +
 +
 +
{| WIDTH="75%"
 +
|-
 +
| style="background-color: #FFFFFF; border: solid 1px #D6D6FF; padding: 1em;" valign="top" |
 +
<center>'''Derivative of the Cosecant'''<br>
 +
<math>\frac{d}{dx}\csc(x)=-\csc(x)\cot(x)</math><br>
 +
</center>
 +
|}
 +
 +
Using the same procedure for the cotangent that we used for the tangent, we get:
 +
 +
 +
{| WIDTH="75%"
 +
|-
 +
| style="background-color: #FFFFFF; border: solid 1px #D6D6FF; padding: 1em;" valign="top" |
 +
<center>'''Derivative of the Cotangent'''<br>
 +
<math>\frac{d}{dx}\cot(x)=-\csc^2(x)</math><br>
 +
</center>
 +
|}
 +
 +
==Resources==
 
* [https://mathresearch.utsa.edu/wikiFiles/MAT1214/Derivatives%20of%20the%20Trigonometric%20Functions/MAT1214-3.5TheDerivativesOfTheTrigonometricFunctionsPwPt.pptx The Derivatives of the Trigonometric Functions] PowerPoint file created by Dr. Sara Shirinkam, UTSA.
 
* [https://mathresearch.utsa.edu/wikiFiles/MAT1214/Derivatives%20of%20the%20Trigonometric%20Functions/MAT1214-3.5TheDerivativesOfTheTrigonometricFunctionsPwPt.pptx The Derivatives of the Trigonometric Functions] PowerPoint file created by Dr. Sara Shirinkam, UTSA.
  
 
* [https://mathresearch.utsa.edu/wikiFiles/MAT1214/Derivatives%20of%20the%20Trigonometric%20Functions/MAT1214-3.5TheDerivativesOfTheTrigonometricFunctionsWS1.pdf The Derivatives of the Trigonometric Functions Worksheet] PowerPoint file created by Dr. Sara Shirinkam, UTSA.
 
* [https://mathresearch.utsa.edu/wikiFiles/MAT1214/Derivatives%20of%20the%20Trigonometric%20Functions/MAT1214-3.5TheDerivativesOfTheTrigonometricFunctionsWS1.pdf The Derivatives of the Trigonometric Functions Worksheet] PowerPoint file created by Dr. Sara Shirinkam, UTSA.
 +
 +
==Licensing==
 +
Content obtained and/or adapted from:
 +
* [https://en.wikibooks.org/wiki/Calculus/Derivatives_of_Trigonometric_Functions Derivatives of Trigonometric Functions, Wikibooks: Calculus] under a CC BY-SA license

Latest revision as of 15:04, 27 October 2021

Sine, cosine, tangent, cosecant, secant, cotangent. These are functions that crop up continuously in mathematics and engineering and have a lot of practical applications. They also appear in more advanced mathematics, particularly when dealing with things such as line integrals with complex numbers and alternate representations of space like spherical and cylindrical coordinate systems.

We use the definition of the derivative, i.e.,

,

to work these first two out.

Let us find the derivative of sin(x), using the above definition.

Definition of derivative
trigonometric identity
factoring
separation of terms
application of limit
solution

Now for the case of cos(x).

Definition of derivative
trigonometric identity
factoring
separation of terms
application of limit
solution

Therefore we have established

Derivative of Sine and Cosine




To find the derivative of the tangent, we just remember that:

which is a quotient. Applying the quotient rule, we get:

Then, remembering that , we simplify:


Derivative of the Tangent


For secants, we again apply the quotient rule.

Leaving us with:

Simplifying, we get:


Derivative of the Secant


Using the same procedure on cosecants:

We get:


Derivative of the Cosecant


Using the same procedure for the cotangent that we used for the tangent, we get:


Derivative of the Cotangent


Resources

Licensing

Content obtained and/or adapted from: