Trigonometry
Trigonometry
Table of Contents
- Angles
- Common Angle Conversion
- Compositions of Trig & Inverse Trig Functions
- Trigonometric Functions
- General Review of Function Transformation
- Trigonometric Functions and the Unit Circle
- Common Trig Values Drill
- Graphing Trigonometric Functions
- Trig graphing exercises
- Graphing Exercises 2
- Trigonometric Identities
- Trigonometric identities: Part I
- Trigonometric identities: Part II
- Trigonometric identities: Part III
Common Angle Conversion
General Review of Function Transformation
[size=200][b][color=#0000ff][u][code][/code]General Function Transformations[br][/u][/color][/b][/size][br]Many functions can be described as [b]transforms[/b] of a [b]parent[/b] function. [br][br]We may perform several transformations on a parent function:[br][br][list][*][b]Vertical Stretch[/b] and [b]Reflection[/b][/*][*][b]Horizontal Stretch [/b]and [b]Reflection[/b][/*][*][b]Horizontal Shift [/b](also called Horizontal Translation)[/*][*][b]Vertical Shift[/b] (also called Vertical Translation)[/*][/list][br]Below we have the parent function [math]\large\sf{f(x)=x^2,}[/math] which will allow us to investigate vertical stretch and reflection and the horizontal and vertical shifts. We will address horizontal stretch using a different function below.[br][br][size=150][b][color=#0000ff][u]I. Vertical Stretch and Reflection[br][/u][/color][/b][/size][br]In the applet,[br][math]\qquad\large{\begin{aligned}\sf{a}&=\textsf{Vertical stretch.}\end{aligned}}[/math][br][br]We see that the child function of the parent with vertical stretch takes the form[br][center][math]\large\sf{g(x)=ax^2.}[/math][/center]________________________________________________________________________________________[br][br][b][color=#3d85c6]Experiment:[/color][br][/b][br]Change the value of [math]\large\sf{a}[/math] and see what happens to the child function.[br][br]Note how [math]\large\sf{a}[/math] governs both vertical stretch [i]and[/i] reflection. Vertical reflection occurs when [math]\large\sf{a}[/math] is a [b]negative[/b] value.[br]
[b]State[/b] your observations regarding the changes in the shape of the quadratic function as [math]\large\sf{a}[/math] changes value.
________________________________________________________________________________________[br][b][u][color=#0000ff][size=150][br]II. Horizontal Shift[/size][/color][/u][/b][br][br]In the applet,[br][math]\qquad\large{\begin{aligned}\sf{a}&=\textsf{Vertical stretch}\\\sf{h}&=\textsf{Horizontal shift.}\end{aligned}}[/math][br][br]We see that the child function with only vertical stretch and horizontal shift is [math]\large\sf{g(x)=a(x-h)^2.}[/math][br][br]Pay attention to how the child function is written. [br][list][*]The variable and the horizontal shift are grouped, [math]\large\sf{\left(x-h\right)}[/math], and then squared. [/*][*]Horizontal shift is [b]subtracted[/b] from [math]\large\sf{x}[/math][/*][/list][br]Example:[br][list][*][math]\large\sf{g(x)=\left(x-4\right)^2}[/math] is a parabola that has been translated to the [b]right[/b] 4 units.[/*][*][math]\large\sf{g(x)=\left(x+3\right)^2}[/math] is a parabola that has been translated to the [b]left[/b] 3 units.[/*][/list][br]________________________________________________________________________________________[br][b][color=#3d85c6][br]Experiment:[br][br][/color][/b]Change the values of [math]\large\sf{a}[/math] and [math]\large\sf{h}[/math] whist taking note of the equation of the child function.
________________________________________________________________________________________[br][b][u][color=#0000ff][size=150][br]III. Vertical Shift[/size][/color][/u][/b][br][br]In the applet,[br][math]\qquad\large{\begin{aligned}\sf{a}&=\textsf{Vertical stretch}\\\sf{h}&=\textsf{Horizontal shift.}\\\sf{k}&=\textsf{Horizontal shift.}\end{aligned}}[/math][br][br]We see that the child function that has vertical stretch and both horizontal and vertical shift is[br][center][math]\large\sf{g(x)=a(x-h)^2+k.}[/math][/center][br][br]For a quadratic, you will notice that [math]\large\sf{\left(h,k\right)}[/math] corresponds to the vertex of the parabola.[br][br]________________________________________________________________________________________[br][br][b][color=#3d85c6]Experiment:[br][br][/color][/b]Change the values of [math]\large\sf{k}[/math] whist taking note of the equation of the child function and the behaviour of the function.[br]
Describe what transformations have been performed on the parent function [math]\large\sf{f(x)=x^2}[/math] to yield[br][center][math]\large\sf{g(x)=\frac{1}{3}x^2-4x+21.}[/math][/center](Yes, algebraic manipulation is needed - remember completing the square?)
________________________________________________________________________________________[br][b][u][color=#0000ff][size=150][br]IV. Horizontal Stretch[/size][/color][/u][/b][br][br]To visualise horizontal stretch, we will look at the function [br][center][math]\large\sf{f(x)=\log_2 x.}[/math][/center][br][br]All the transformations discussed previously apply, [br][math]\qquad\large{\begin{aligned}\sf{a}&=\textsf{Vertical stretch}\\\sf{h}&=\textsf{Horizontal shift.}\\\sf{k}&=\textsf{Horizontal shift,}\end{aligned}}[/math][br]which we can see applied to the child function[br][center][math]\large\sf{g(x)=a\cdot\log_2\left(bx-h\right)+k.}[/math][/center][br][br]In addition, we have horizontal stretch/compression and reflection given by [math]\large\sf{b.}[/math][br][br]________________________________________________________________________________________[br][br][b][color=#3d85c6]Experiment:[br][br][/color][/b]Set the values for [math]\large\sf{a}[/math] to 1 and [math]\large\sf{h}[/math] to 0. Change the value of [math]\large\sf{b}[/math] to see how the asymptote at x=0 does not change.[br][br]Compare the behaviour of the function as [math]\large\sf{a}[/math] changes to that as [math]\large\sf{b}[/math]. They are different! The most striking difference occurs when these coefficients change sign.
Describe the difference that you see in the graph when you change horizontal stretch compared to vertical stretch for the transformation of the logarithmic function. [br][br]What happens when either of these values turns negative?
________________________________________________________________________________________[br][b][u][color=#0000ff][size=150][br]V. Summary[/size][/color][/u][/b][br][br]In this activity, we have reviewed the main transformation that may be performed on a parent function:[br][list][*]Vertical stretch and reflection,[/*][*]Horizontal stretch and reflection,[/*][*]Vertical shift, and[/*][*]Horizontal shift[/*][/list]________________________________________________________________________________________[br]
[b]Explain[/b] how the examination of a function's graph allows you to discern which transformations a parent function has undergone to yield its child.[br][br][b]Use examples[/b] during your discussion.
Trigonometric identities: Part I
[size=200][b][size=150]Recall the following reciprocal, co-function, even-odd identities as well as the Pythagorean identities.[br][/size][/b][/size][size=150]Whilst you are learning the identities, write the full equation for the identity.[/size][br][br][i][color=#5b0f00][b]Example:[/b][/color][br][br][/i][b]Question:[/b][br] Double angle identity:[br] [math]\sin\left(2x\right)=[/math][br][br][b]Answer:[/b][br] [math]\sin\left(2x\right)=2\sin\left(x\right)\cos\left(x\right)[/math]
What is the pythagorean identity that uses [math]\sin^2\left(x\right)[/math]?
Reciprocal identity:[br][math]\frac{1}{\cos\left(x\right)}=[/math]
Cofunction identity:[br][math]\tan\left(x\right)=[/math]
Cofunction identity:[br] [math]\cos\left(x\right)=[/math]
Reciprocal identity:[br][math]\frac{1}{\sin\left(x\right)}=[/math]
Even-odd identity:[br][math]\sec\left(-x\right)=[/math]
Cofunction identity:[br][math]\sin\left(\frac{\pi}{2}-x\right)=[/math]
Reciprocal identity:[br][math]\frac{1}{\sec\left(x\right)}=[/math]
Cofunction identity:[br][math]\sec\left(\frac{\pi}{2}-x\right)=[/math]
Even-Odd Identity:[br][math]\sin\left(-x\right)=[/math]
What is the pythagorean identity that uses [math]\tan^2\left(x\right)[/math]?
Even-odd identity:[br][math]\tan\left(-x\right)=[/math]
Reciprocal identity:[br][math]\frac{1}{\csc\left(x\right)}=[/math]
Reciprocal identity:[br][math]\cos\left(x\right)=[/math]
Cofunction identity:[br][math]\cot\left(\frac{\pi}{2}-x\right)=[/math]
Reciprocal identity:[br] [math]\cot\left(x\right)=[/math]
Reciprocal Identity:[br][math]\csc\left(x\right)=[/math]
Reciprocal identity:[br][math]\frac{1}{\tan\left(x\right)}=[/math]
Cofunction identity:[br][math]\cot\left(x\right)=[/math]
Cofunction identity:[br][math]\sin\left(x\right)=[/math]
Reciprocal identity:[br][math]\sec\left(x\right)=[/math]
What are the three ways one can write the pythagorean identity using trigonometric functions?
Reciprocal identity[br][math]\tan\left(x\right)=[/math]
Cofunction identity:[br][math]\csc\left(x\right)=[/math]
Cofunction identity:[br][math]\tan\left(\frac{\pi}{2}-x\right)=[/math]
Even-odd identity:[br][math]\cot\left(-x\right)=[/math]
Reciprocal identity:[br] [math]\frac{1}{\cot\left(x\right)}=[/math]
Even-Odd Identity:[br][math]-\sin\left(x\right)=[/math]
Cofunction identity[br][math]\cos\left(\frac{\pi}{2}-x\right)=[/math]
Cofunction identity:[br][math]\sec\left(x\right)=[/math]
Reciprocal identity:[br][math]\sin\left(x\right)=[/math]
Cofunction identity:[br][math]\csc\left(\frac{\pi}{2}-x\right)=[/math]
Even-odd identity:[br][math]\cos\left(-x\right)=[/math]
What is the pythagorean identity that uses [math]\csc^2\left(x\right)[/math]?