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

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 ratios^{1}.

Angles (Degrees) | Angles (Radians) | ||||||
---|---|---|---|---|---|---|---|

0 | |||||||

**NOTE**: On all Assessments, **exact** trigonometric values are expected, except where instructed otherwise. There are two options available for evaluating trigonometric ratios, e.g.,

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

Evaluate the following:

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

**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*requires ∗ for explicit multiplication, e.g., is$4\pi $ .$4\text{*pi}$ - To execute code, press the “View the result” button:

**Half-Angle Formulas**

**Double Angle Formulas**

**Mathematica**

**NOTE**: On * most* Assessments,

`x1``(* 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.

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. ↩