---
title: "Demo 11"
author: 
  - name: "Kwang Hyun Kim"
    email: "kkim@qcc.cuny.edu"
    affiliation: "Queensborough Community College"
date: 12/02/2022
execute:
  echo: false
slide-format: revealjs
format:
  revealjs: 
    keep-md: false # Debugging
    keep-ipynb: false # Debugging
    slide-number: true
    css: [lib/css/details.css,lib/css/extra_reveal.css]
    embed-resources: true
    standalone: true # One file
    controls: true
    preview-links: true
    link-external-newwindow: true
    html-math-method:
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      url: "https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"
    transition: none
    theme: simple
    background-transition: none
    scrollable: true
    #width: 1920
    height: 1080
    margin: 0.05
---
## Right Triangle and Trigonometry

::: {layout-ncol=2}
By Pythagorean theorem, $$x^2+y^2=r^2$$

![](lib/pic/right_triangle.svg){width="250px"}
:::
::: {layout-ncol=3}
$$\sin\theta=\frac{y}{r}$$
$$\cos\theta=\frac{x}{r}$$

$$\tan\theta=\frac{\sin\theta}{\cos\theta}=\frac{y}{x}$$
$$\csc\theta=\frac{1}{\sin\theta}=\frac{r}{y}$$

$$\sec\theta=\frac{1}{\cos\theta}=\frac{r}{x}$$
$$\cos\theta=\frac{1}{\tan\theta}=\frac{x}{y}$$
:::

## Practice 1: No Decimal approximation!
:::{layout="[3,2]"}
Find the exact value of all six trigonometric functions of the angle $A$. 

![](lib/pic/p1.svg){width="120%"}
:::
<details>
<summary>Show</summary> 
::: {.panel-tabset}
## $r$
$$
r=\sqrt{\left(12\right)^2+\left(-5\right)^2}=\sqrt{169}=13$$

## $\sin,\csc$
$$
\sin A=\frac{y}{r}={\color{red}-\frac{5}{13}},\csc A={\color{red}-\frac{13}{5}}
$$

## $\cos,\sec$
$$\cos A=\frac{x}{r}={\color{red}\frac{12}{13}},~
\sec A={\color{red}\frac{13}{12}}$$

## $\tan,\cot$
$$\tan A=\frac{x}{y}={\color{red}-\frac{5}{12}},~
\cot A={\color{red}-\frac{12}{5}}$$
:::
</details>

## Practice 2: Exact values with a ref angle

Determine the exact value of $\tan 120^\circ$ . (Decimal approximations will **NOT** be accepted.)

<details>
<summary>Show</summary>
:::{layout="[5,2]"}
$120^\circ$ is in Quadrant II. <br/>
- $\tan(120^\circ) < 0$ <br/>
- $ref(120^\circ)=180^\circ-120^\circ= 60^\circ$
$$\tan 120^\circ=-\tan(60^\circ)=-\sqrt{3}$$

![](lib/pic/p2.svg)
:::
</details>