Fundamental Theorem of Calculus (FTC)

Fundamental Theorem of Calculus (FTC)Prerequisite KnowledgeTrigonometryEvaluating Trigonometric RatiosPractice 02Use Technology to Rewrite Using AlgebraRewriting Trigonometric ExpressionsUse Technology to Rewrite Using Trigonometric IdentitiesCC BY-NC-SA 4.0

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

Section 4.9 – Antiderivatives
Section 5.3 – The Fundamental Theorem of Calculus
Section 5.4 – Indefinite Integrals and the Net Change Theorem

Prerequisite Knowledge

Trigonometry

Evaluating Trigonometric Ratios

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.

Angles (Degrees)	Angles (Radians)	$\sin{(\theta)}$	$\cos{(\theta)}$	$\tan{(\theta)}=\frac{\sin{(\theta)}}{\cos{(\theta)}}$	$\cot{(\theta)}=\frac{1}{\tan{(\theta)}}$	$\csc{(\theta)}=\frac{1}{\sin{(\theta)}}$	$\sec{(\theta)}$
$0^{\circ}$	0	$\frac{\sqrt{0}}{2}=0$	$\frac{\sqrt{4}}{2}=1$	$\frac{\sqrt{0}}{\sqrt{4}}=0$	$\frac{\sqrt{4}}{\sqrt{0}}=\text{undef.}$	$\frac{2}{\sqrt{0}}=\text{undef.}$	$\frac{2}{\sqrt{4}}=1$
$30^{\circ}$	$\frac{\pi}{6}$	$\frac{\sqrt{1}}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{1}}{\sqrt{3}}$	$\frac{\sqrt{3}}{\sqrt{1}}$	$\frac{2}{\sqrt{1}}$	$\frac{2}{\sqrt{3}}$
$45^{\circ}$	$\frac{\pi}{4}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{\sqrt{2}}=1$	$\frac{\sqrt{2}}{\sqrt{2}}=1$	$\frac{2}{\sqrt{2}}$	$\frac{2}{\sqrt{2}}$
$60^{\circ}$	$\frac{\pi}{3}$	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{1}}{2}$	$\frac{\sqrt{3}}{\sqrt{1}}$	$\frac{\sqrt{1}}{\sqrt{3}}$	$\frac{2}{\sqrt{3}}$	$\frac{2}{\sqrt{1}}$
$90^{\circ}$	$\frac{\pi}{2}$	$\frac{\sqrt{4}}{2}=1$	$\frac{\sqrt{0}}{2}=0$	$\frac{\sqrt{4}}{\sqrt{0}}=\text{undef.}$	$\frac{\sqrt{0}}{\sqrt{4}}=0$	$\frac{2}{\sqrt{4}}=1$	$\frac{2}{\sqrt{0}}=\text{undef.}$

NOTEexact $\sin{\left(\frac{4\pi}{3}\right)}$ :

$\sin{\left(\frac{4\pi}{3}\right)}$ ; or
$\sin{\left(\frac{4\pi}{3}\right)}$ $-\frac{\sqrt{3}}{2}$

Practice 02

Evaluate the following:

$\sin{\left(\frac{4\pi}{3}\right)}$
$\sec{\left(\frac{5\pi}{6}\right)}$
$\tan{\left(\frac{3\pi}{2}\right)}$
$\csc{\left(\frac{\pi}{6}\right)}$
$\cos{\left(\frac{11\pi}{6}\right)}$
$\cot{\left(\frac{3\pi}{4}\right)}$
$\cos{\left(\frac{5\pi}{3}\right)}$
$\sin{\left(\frac{2\pi}{3}\right)}$
$\cos{\left(\pi\right)}$
$\tan{\left(\frac{7\pi}{6}\right)}$

Use Technology to Rewrite Using Algebra

Mathematica

NOTE: On all Assessments, Mathematica may be used to evaluate any trigonometric expression without showing any work.


1
(* Evaluate: sin(4 pi/3) *)
2
Sin[4Pi/3]

Warnings:

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.
Remember, correct Mathematica code will be all black except for variables.
To execute code (including comment codes), press and hold the SHIFT key and press the ENTER key.

Python

Warnings:

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 Python syntax is located at: https://www.jjw3.com/Common_Python_Code.html.
Python $4\pi$ $4\text{*pi}$ .
To execute code, press the “View the result” button:

Rewriting Trigonometric Expressions

Half-Angle Formulas

$\displaystyle \cos^2{\!(x)}=\frac{1}{2}\left(1+\cos{(2x)}\right)$

$\displaystyle \sin^2{\!(x)}=\frac{1}{2}\left(1-\cos{(2x)}\right)$

Double Angle Formulas

$\displaystyle \sin{(2x)}=2\sin{(x)}\cos{(x)}$

$\displaystyle \cos{(2x)}=\cos^2{\!(x)}-\sin^2{\!(x)}$

Use Technology to Rewrite Using Trigonometric Identities

Mathematica

NOTE: On most Assessments, Mathematica may be used to rewrite any trigonometric expression without showing any work.


x
1
(* Rewrite: sin(x)^2 *)
2
TrigReduce[Sin[x]^2]
3

4
(* Rewrite: sin(2x) *)
5
TrigExpand[Sin[2x]]

Warnings:

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.
Remember, correct Mathematica code will be all black except for variables.
To execute code (including comment codes), press and hold the SHIFT key and press the ENTER key.

CC BY-NC-SA 4.0

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License [http://creativecommons.org/licenses/by-nc-sa/4.0/].

Created: Monday, 18 January 2021 09:33 EDT Last Modified: Monday, 23 May 2022 - 01:54 (EDT)

1 I did not take trigonometry in high school. In Calculus I during my first semester of college, my professor showed us this table – I was hooked, I recognized that Mathematics was simply finding patterns, and changed my major to Mathematics from Chemistry – I eventually changed it to a dual major in Mathematics and Chemistry. ↩