Trigonometric Integrals

Author: John J Weber III, PhD Corresponding Textbook Sections:

• Section 7.2 – Trigonometric Integrals

Trigonometric Integrals

Prerequisite Knowledge

Trigonometric Identities

The Pythagorean Identities shown in red will be needed for integrals of trigonometric functions.

Pythagorean Identity

One Pythagorean Identity that you should remember from Pre-Calculus (Math 1113):

${\sin }^2(x)+{\cos }^2(x)=1$$\sin^2{(x)}+\cos^2{(x)}=1$

When integrating trigonometric integrals, you will need two equivalent forms of the above Pythagorean Identity:

• ${\sin }^2(x)=1-{\cos }^2(x)$$\color{red}\sin^2{(x)}=1-\cos^2{(x)}$
• ${\cos }^2(x)=1-{\sin }^2(x)$$\color{red}\cos^2{(x)}=1-\sin^2{(x)}$

Additional Pythagorean Identities

Pythagorean Identity with Tangent and Secant

${\tan }^2(x)+1={\sec }^2(x)$$\tan^2{(x)}+1=\sec^2{(x)}$

Derivation

NOTE: You are not expected to derive the Pythagorean Identity for tangent and secant; however, if you forget the Pythagorean Identity for tangent and secant, then knowing how to derive the identity will be useful.

${\sin }^2(x)+{\cos }^2(x)=1$$\sin^2{(x)}+\cos^2{(x)} = 1$

Divide through by ${\displaystyle {\cos }^2(x)}$$\displaystyle \cos^2{(x)}$.

$\Rightarrow {\displaystyle \frac{{\sin }^2(x)}{{\cos }^2(x)}}+{\displaystyle \frac{{\cos }^2(x)}{{\cos }^2(x)}}={\displaystyle \frac{1}{{\cos }^2(x)}}$$\Rightarrow \dfrac{\sin^2{(x)}}{\cos^2{(x)}} + \dfrac{\cos^2{(x)}}{\cos^2{(x)}} = \dfrac{1}{\cos^2{(x)}}$

$\Rightarrow {\tan }^2(x)+1={\sec }^2(x)$$\Rightarrow \color{red}\tan^2{(x)} + 1 = \sec^2{(x)}$

When integrating trigonometric integrals, you will need the equivalent form of the above Pythagorean Identity:

${\tan }^2(x)={\sec }^2(x)-1$$\color{red}\tan^2{(x)}=\sec^2{(x)}-1$

Pythagorean Identity with Cotangent and Cosecant

$1+{\cot }^2(x)={\csc }^2(x)$$1+\cot^2{(x)}=\csc^2{(x)}$

Derivation

NOTE: You are not expected to derive the Pythagorean Identity for cotangent and cosecant; however, if you forget the Pythagorean Identity for cotangent and cosecant, then knowing how to derive the identity will be useful.

${\sin }^2(x)+{\cos }^2(x)=1$$\sin^2{(x)}+\cos^2{(x)} = 1$

Divide through by ${\displaystyle {\sin }^2(x)}$$\displaystyle \sin^2{(x)}$.

$\Rightarrow {\displaystyle \frac{{\sin }^2(x)}{{\sin }^2(x)}}+{\displaystyle \frac{{\cos }^2(x)}{{\sin }^2(x)}}={\displaystyle \frac{1}{{\sin }^2(x)}}$$\Rightarrow \dfrac{\sin^2{(x)}}{\sin^2{(x)}} + \dfrac{\cos^2{(x)}}{\sin^2{(x)}} = \dfrac{1}{\sin^2{(x)}}$

$\Rightarrow 1+{\cot }^2(x)={\csc }^2(x)$$\Rightarrow \color{red}1 + \cot^2{(x)} = \csc^2{(x)}$

When integrating trigonometric integrals, you will need the equivalent form of the above Pythagorean Identity:

${\cot }^2(x)={\csc }^2(x)-1$$\color{red}\cot^2{(x)}=\csc^2{(x)}-1$

Half-angle formulas

NOTE: You will need to appropriately use the following trigonometric identities. You are not expected to memorize these identities.

• ${\displaystyle {\sin }^2(x)=\frac{1-\cos (2x)}{2}}$$\displaystyle \color{blue}\sin^2{\!(x)}=\frac{1-\cos{(2x)}}{2}$
• ${\displaystyle {\cos }^2(x)=\frac{1+\cos (2x)}{2}}$$\displaystyle \color{blue}\cos^2{\!(x)}=\frac{1+\cos{(2x)}}{2}$

Use Technology to Recall Half-Angle Formulas

Mathematica

NOTE: On all Assessments, Mathematica may be used to appropriately rewrite an even power of sine or cosine without showing any work.

1
(* Rewrite sin^2(x) *)
2
TrigReduce[Sin[x]^2]

Warnings:

1. Be very careful with the syntax. Syntax is the set of rules on how to write computer code. Every software program has its own unique syntax. Some basic Mathematica syntax is located at: https://www.jjw3.com/Common_Mathematica_Code.html.

2. TrigReduce[ ] has at least one argument:

   1. The trigonometric expression to be rewritten.

3. For help on using the TrigReduce[ ] function:

   1. In Mathematica, execute the code: $\text{?TrigReduce}$$\text{?TrigReduce}$
   2. Click on $\vee$$\or$ near the bottom-left of output
   3. Click on local
   4. Read how to use the TrigReduce[ ] function – you will be able to copy-paste code.

4. Remember, correct Mathematica code will be all black except for variables.

5. To execute code (including comment codes), press and hold the SHIFT key and press the ENTER key.

6. Mathematica may return a result in a different, but equivalent, form than your answer. In these cases, use FTC-I to check your work.

Double-Angle Formulas

NOTE: You will need to appropriately use the following trigonometric identities. You are not expected to memorize these identities.

1. $\sin (2x)=2\sin (x)\cos (x)$$\color{purple}\sin{(2x)}=2\sin{(x)}\cos{(x)}$
2. $\cos (2x)={\cos }^2(x)-{\sin }^2(x)$$\color{purple}\cos{(2x)}=\cos^2{(x)}-\sin^2{(x)}$

Check Your Work

Use Technology to Recall Double-Angle Formulas

Mathematica

NOTE: On all Assessments, Mathematica may be used to appropriately rewrite an even power of sine or cosine without showing any work.

xxxxxxxxxx
2
1
(* Rewrite sin(2x) *)
2
TrigExpand[Sin[2x]]

Warnings:

1. Be very careful with the syntax. Syntax is the set of rules on how to write computer code. Every software program has its own unique syntax. Some basic Mathematica syntax is located at: https://www.jjw3.com/Common_Mathematica_Code.html.

2. TrigExpand[ ] has at least one argument:

   1. The trigonometric expression to be rewritten.

3. For help on using the TrigExpand[ ] function:

   1. In Mathematica, execute the code: $\text{?TrigExpand}$$\text{?TrigExpand}$
   2. Click on $\vee$$\or$ near the bottom-left of output
   3. Click on local
   4. Read how to use the TrigExpand[ ] function – you will be able to copy-paste code.

4. Remember, correct Mathematica code will be all black except for variables.

5. To execute code (including comment codes), press and hold the SHIFT key and press the ENTER key.

6. Mathematica may return a result in a different, but equivalent, form than your answer. In these cases, use FTC-I to check your work.

NOTE: There is NO reason to memorize when to use TrigReduce[ ] and TrigExpand[ ]; one will return the same input and the other will rewrite the input.

Evaluation of Trig Functions

There is NO reason to memorize the unit circle. Look at the Table below and consider the patterns of the numbers. Use these patterns to recall your trig ratios¹.

Evaluating Trigonometric Ratios

NOTE: On all Assessments, exact trigonometric values are expected, except where instructed otherwise. There are two options available for evaluating trigonometric ratios, e.g., $\sin \left(\frac{4\pi }{3}\right)$$\sin{\left(\frac{4\pi}{3}\right)}$:

1. Leave the answer as $\sin \left(\frac{4\pi }{3}\right)$$\sin{\left(\frac{4\pi}{3}\right)}$; or
2. Rewrite $\sin \left(\frac{4\pi }{3}\right)$$\sin{\left(\frac{4\pi}{3}\right)}$ as $-\frac{\sqrt{3}}{2}$$-\frac{\sqrt{3}}{2}$

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Created: Monday, 18 January 2021 09:14 EDT Last Modified: Monday, 10 January 2022 - 07:17 (EST)