Lila: /* Resources */

2021-10-27T21:04:30Z

Resources

Khanh at 17:26, 10 October 2021

2021-10-10T17:26:59Z

James.kercheville: Added links for PowerPoint and worksheet

2020-08-13T23:14:05Z

Added links for PowerPoint and worksheet

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* [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.

← Older revision		Revision as of 21:04, 27 October 2021
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	* [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

← Older revision		Revision as of 17:26, 10 October 2021
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		+
		+	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.

Derivatives of the Trigonometric Functions - Revision history

Lila: /* Resources */

Khanh at 17:26, 10 October 2021

James.kercheville: Added links for PowerPoint and worksheet