Fundamental Math
IT100 Session 10: Trigonomic Functions
1 Nov 2017
Agenda
Agenda
- Solving Trigonometric Equations with Identities
- Sum and Difference Identities
- Double-Angle, Half-Angle, and Reduction Formulas
- Sum-to-Product and Product-to-Sum Formulas
- Solving Trigonometric Equations
- Modeling with Trigonometric Equations
Your petition
Water Bottle Bungee Challenge
Mapping
\[ \begin{array}{lrrr} time & Angle\ A & Angle\ B & Angle\ C \\ 9:10:00 & 58 & 4 & 118\\ 9:10:10 & 15 & 1 & 164\\ 9:10:20 & -38 & -2 & -140\\ 9:10:30 & -60 & -5 & -115\\ 9:10:40 & -62 & -8 & -110\\ 9:10:50 & -68 & -11 & -101\\ 9:11:00 & -71 & -14 & -95\\ 9:11:20 & -73 & -17 & -90\\ 9:11:30 & -75 & -20 & -85\\ 9:11:40 & -77 & -23 & -80\\ 9:11:50 & -79 & -26 & -75\\ 9:12:00 & -81 & -29 & -70\\ \end{array} \]
Line of Sight Distances
\[ \begin{array}{rrrr} time & BC & AC & AB \\ \hline 0 & 3.71 & 0.31 & 3.97 \\ 10 & 3.72 & 0.25 & 3.97 \\ 20 & 3.73 & 0.22 & 3.97 \\ 30 & 3.74 & 0.38 & 3.97 \\ 40 & 3.74 & 0.59 & 3.97 \\ 50 & 3.75 & 0.77 & 3.97 \\ 60 & 3.77 & 0.96 & 3.97 \\ 70 & 3.80 & 1.16 & 3.97 \\ 80 & 3.85 & 1.36 & 3.97 \\ 90 & 3.93 & 1.58 & 3.97 \\ 100 & 4.03 & 1.80 & 3.97 \\ 110 & 4.17 & 2.05 & 3.97 \\ \end{array} \]
Height
\[ \begin{array}{lrrr} Time & Angle & BC & Height\\ 0& 2 & 0.13 &133 \\ 10& 6 & 0.39 &390 \\ 20& 10 & 0.66 &660 \\ 30&14 & 0.91 &914 \\ 40&18 & 1.15 &1,153 \\ 50&21 & 1.34 & 1,344 \\ 60&24 & 1.53 &1,533 \\ 70&26 & 1.66 &1,664 \\ 80&28 & 1.81 &1,807 \\ 90&30 & 1.96 &1,964 \\ 100&30 & 2.02 & 2,017 \\ 110&30 & 2.09 & 2,086 \\ \end{array} \]
Trigonometric Equations and Identities
Basic definitions
\[ \large\begin{array}{rrcll} cosine: & \cos\ \theta &=&{x \over r} \\ sine: & \sin\ \theta &=&{y \over r} \\ tangent: & \tan\ \theta &=& {y \over x} \\ secant: & \sec\ \theta &=& {r \over x} \\ cosecant: &\csc\ \theta &=& {r \over y} \\ cotangent: &\cot\ \theta &=& {y \over x} \\ \end{array} \]
Graphs
Identities
\[ \large\begin{array}{rcl} \sin\theta &=& \cos\left({\pi\over 2} - \theta\right) = \cos \left(90 -\theta\right)\\ \cos\theta &=& \sin\left({\pi\over 2} - \theta\right) = \sin \left(90 -\theta\right)\\ \tan\theta &=& \cot\left({\pi\over 2} - \theta\right) = \tan \left(90 -\theta\right)\\ \sin\left(\theta\right)^2 + \cos\left(\theta\right)^2 & = & 1\\ \end{array} \]
Half-Angle, Double-Angle, and Reduction Formulas
Half angle formulas
\[ \large\begin{array}{rcl} \sin\left({\theta\over 2}\right) & = & \pm \sqrt{1- \cos(\theta)\over 2}\\ \\ \cos\left({\theta\over 2}\right) & = & \pm \sqrt{1 + \cos(\theta)\over 2}\\ \\ \tan\left({\theta\over 2}\right) & = & \pm \sqrt{1 - \cos(\theta)\over 1 + \cos(\theta)} = {\sin(\theta) \over 1 + cos(\theta)} = {1 - cos(\theta)\over \sin(\theta)}\\ \end{array} \]
Double angle formulas
\[ \large\begin{array}{rcl} \sin\left(2\theta\right) & = & 2 \sin(\theta) \cos(\theta)\\ \\ \cos\left(2\theta\right) & = & \cos^2(\theta) - \sin^2(\theta) = 1 - 2 \sin^2(\theta) = 2 \cos^2(\theta) - 1\\ \\ \tan\left(2\theta\right) & = & {2 \tan(\theta)\over 1 - \tan^2(\theta)} \\ \end{array} \]
Reduction Formula
\[ \large\begin{array}{rcl} \sin^2 \theta & = & {1 - \cos(2\theta)\over 2}\\ \\ \cos^2 \theta & = & {1 + \cos(2\theta)\over 2}\\ \\ \tan^2 \theta & = & {1 - \cos(2\theta)\over 1 + \cos(2\theta)}\\ \end{array} \]
Sum, Difference, Product
Sum Formulas
\[ \large\begin{array}{rcl} \sin(\alpha + \beta) & = & \sin\alpha \cos\beta + \cos\alpha \sin\beta\\ \\ \cos(\alpha + \beta) & = & \cos\alpha \cos\beta - \sin\alpha \sin\beta\\ \\ \tan(\alpha + \beta) & = & {\tan\alpha + \tan\beta\over 1 - \tan\alpha \tan\beta}\\ \end{array} \]
Difference Functions
Difference \[ \large\begin{array}{rcl} \sin(\alpha - \beta) & = & \sin\alpha \cos\beta - \cos\alpha \sin\beta\\ \\ \cos(\alpha - \beta) & = & \cos\alpha \cos\beta + \sin\alpha \sin\beta\\ \\ \tan(\alpha - \beta) & = & {\tan\alpha - \tan\beta\over 1 + \tan\alpha \tan\beta}\\ \end{array} \]
Product to Sum
\[ \large\begin{array}{rcl} \cos\alpha\ \cos \beta & = & {\cos(\alpha -\beta) + \cos(\alpha + \beta)\over 2}\\ \\ \sin\alpha\ \sin \beta & = & {\cos(\alpha -\beta) - \cos(\alpha + \beta)\over 2}\\ \\ \sin\alpha\ \cos \beta & = & {\sin(\alpha +\beta) + \sin(\alpha - \beta)\over 2}\\ \\ \cos\alpha\ \sin \beta & = & {\sin(\alpha +\beta) - \sin(\alpha - \beta)\over 2}\\ \end{array} \]
Law of Sines
\[ \large {\sin \alpha\over a} = {\sin \beta\over b} = {\sin \gamma \over c} \]
\[ \large {a \over \sin \alpha} = {b \over \sin \beta} = {c \over \sin \gamma} \]
Area of an oblique triangle
\[ \large\begin{array}{rcl} Area &=& {bc\ \sin \alpha\over 2}\\ &=& {ac\ \sin \beta\over 2}\\ &=& {ab\ \sin \gamma\over 2}\\ \end{array} \]
requires 2 sides and the angle between them
Heron's Formula for Area
Requires the lengths of all sides
\[ \large\begin{array}{rcl} Area & = & \sqrt{s (s-a) (s-b) (s-c)}\\ \\ where\\ \\ s & = & {a + b + c\over 2}\\ \end{array} \]
Linear systems
- Day 1: 120 baht
- 3 coffee
- 2 bottled tea
- Day 2: 75 baht
- 2 coffee
- 1 bottled tea
Equation:
\[ \begin{array}{rcl} 3c + 2t &=& 120 \\ 2c + 1t &=& 75 \\ \end{array} \]
Matrix representation:
\[ \begin{array}{ccc} 3 & 2 & 120 \\ 2 & 1 & 75 \\ \end{array} \]
Gaussian Elimination
- Divide the first row by the value in the first column
- Use this adjusted row to clear the first column in subsequent rows
- Repeat for each subsequent column
\[ \left[\begin{array}{ccc} 3 & 2 & 120 \\ 2 & 1 & 75 \\ \end{array}\right]\Rightarrow \left[\begin{array}{ccc} 1 & 0.67 & 40 \\ 0 & -0.33 & -5 \\ \end{array}\right]\Rightarrow \left[\begin{array}{ccc} 1 & 0.67 & 40 \\ 0 & 1 & 15 \\ \end{array}\right] \]
Gaussian Elimination: Worked examples
\[ \begin{array}{ccc} 3 & 2 & 120 \\ 2 & 1 & 75 \\ \\ 1 & 2/3 & 40 \\ 2 & 1 & 75 \\ \\ 1 & 2/3 & 40 \\ 2-1\times 2 & 1- 2 \times 2/3 & 75 - 2\times 40 \\ \\ 1 & 2/3 & 40 \\ 0 & -1/3 & -5\\ \\ 1 & 2/3 & 40 \\ 0 & 1 & 15 \\ \end{array} \]
Back substitution
- Use the values in the bottom row to remove the values in the column above.
- Repeat until there is an identity matrix
- The solution is the identity matrix
\[ \left[\begin{array}{ccc} 1 & 2/3 & 40 \\ 0 & 1 & 15 \\ \end{array}\right]\Rightarrow \left[\begin{array}{ccc} 1 & 0 & 30 \\ 0 & 1 & 15 \\ \end{array}\right] \]
Back substitution: example
\[ \begin{array}{ccc} 1 & 0.67 & 40 \\ 0 & 1 & 15 \\ \\ 1 & 2/3 - 2/3 & 40 - 15*2/3\\ 0 & 1 & 15 \\ \\ 1 & 0 & 30 \\ 0 & 1 & 15 \\ \end{array} \]