Happy Spring Break!

Hope everyone is having a wonderful spring break! Remember to finish up your yellow and purple review booklets. I will be posting the answers on the weekend. Remember to post any questions or problems that you may require some help with. Your classmates may be able to give you a hand with these.

March 23, 2011 More on Identities

Identities Lesson 2

This might also help a bit.....

March 22, 2011

I have attached today's first lesson on solving Trig Identities. 

Identities Lesson 1

Remember to always pick one side and "work it out" to look like the other side.  Many of these solutions come out quite easily if you express everything on the most complex side in terms of sine and cosine only.  There can be various ways to arrive at the same solution so don't despair if your neighbor has completed the solution in a different way.

Example 1: 

Prove that

sin y + sin y cot² y = csc y :         SOLUTION BELOW

Example 2: 

Prove that

sin x cos x tan x = 1 − cos² x:    SOLUTION BELOW

Example 3:

Prove that

     SOLUTION BELOW

The handout sheet of Identities is due to hand in on Friday March 25th, 2011.  Remember you only need to complete the ODD number questions.   Please hand in at the end of class.

Trigonometric Identities

Moving on to the third unit, we discussed a very challenging lesson on how to prove Trigonometric Identities.

First, we did a recap about the eight trigonometric identities (Pythagorean and Reciprocal)

Pythagorean/Main Identities:

☻cos²θ + sin²θ = 1 ---> dividing both sides by cos²θ and by sin²θ, you'll get:

☻1 + tan²θ = sec²θ and ☻1 + cot²θ = csc²θ respectively.

Reciprocal/Basic Identities:

☻sinθ = 1/cscθ ☻cscθ = 1/sinθ

☻cosθ = 1/secθ ☻secθ = 1/cosθ

☻tanθ = 1/cotθ ☻cotθ = 1/tanθ

Next, in order to prove equations using identities, you must:

1. Use basic identity substitution.

2. Change everything to sine or cosine

3. Simplify all complex fractions into simple fraction.

4. Use fraction rules for addition and subtraction.

5. Factor if exponents are not the same on both sides.

Lastly, here are some important things to consider in this unit:

1. Familiarize yourself with all the 8 trig. identities. The better you know them, the easier it will be to recognize what is going on in the problems.

2. Work on the most complex side and simplify it so that it has the same form as the simplest side.

3. Don't work on both sides of the equation and try to meet in the middle. Start on one side and make it look like the other side.

4. In most examples where you see power 2 (that is, ²), it will involve using the Pythagorean identities or their derivatives.

Feel free to post a comment if there are points or clarifications you might want to add!

Carpe diem!

Model and Solve Problems using Trig Functions - March 21, 2011

We have finished up with Unit B on Transformations.  The last two lessons are attached in the links.  Enjoy!

Transformations Lesson 5

Transformations Lesson 6

March 17,2011

What's coming up???

A busy week just around the corner....
Week before spring break...Yahoooo!!!

On Monday we will finish up with our last lesson in the Transformations Unit.  Exercise 13 lesson looking at some real life context situations for using sinusoidal equations.

How in the world is this Ferris Wheel connected to sinusoidal equations???

Stay tuned...

Tuesday and Wednesday we will get into our next unit on Identities.  Look interesting?? 

Thursday and Friday will be working days on both Unit A and Unit B Review.  
Over spring break you will have two review booklets to complete in preparation for the unit assessments coming up. 
 
If you are having trouble with any of the review work, post a blog about the specific question.  One of your classmates may be able to help you out with the solution.  I will post the answers to the booklets on Sunday April 3rd so that you can check your work just in time for getting back on Monday.

Be sure to visit the blog!

We will continue on with Identities when we get  back on Monday April 4th.


Transformations Unit Assessment - Tuesday April 12th,2011.

Identities Unit Assessment - Thursday April 14th, 2011. 

Graphing Reciprocal Functions, Absolute Values and Trig Reciprocals Review

(March 16,2011)

Todays class was extremely productive. We covered almost three exercises in under two hours. (YAY!) In the morning, we finished off the notes from exercise 10(graphing reciprocal functions), and Graphing Absolute Values(exercise 11) and in the afternoon, got right into Graphing Trig Reciprocals. Summary of todays notes will be posted down below so keep reading. =)

Exercise 10: Graphing Reciprocal Functions

Steps in graphing a reciprocal function:

• First of all, make sure you use the reciprocal of the function instead of the original one.

• Next, you have to find the asymptote. To do this you have take the denominator (-x/2-2) and set y to 0:
• ( 0=-x/2-2 and solve for "x")

• Once you have "x" you can then draw in your asymptote with a dotted line (in this case x=-4, so our asymptote=-4
• Then find the invariant points(turn points). ***Remember in reciprocal graphs, the invariant points are located at y=1 & y=-1 (Again take the denominator and set y to -1 and 1) 1=-x/2-2 and -1=-x/2-2 then solve for "x".
• Next plot these points and graph the reciprocal function

Exercise 11: Graphing Absolute Values

The easiest way to graph absolute values is to make a table of values.

- But remember that the absolute value of a function is always positive or zero. So all the points that are below the x-axis(-ve points) get reflected to above the x-axis.

Your new table of values should look like

this:

- Next just plot the points in the graph and connect the dots.

NO NEGATIVE "Y" VALUES

Graphing Trig Reciprocals:

Example:Graph a Cosecant Graph (We all know from the previous unit that Cosecant is the reciprocal of sin. (Cosecant= 1/sinx)

-Start off by graphing your typical y=sinx graph

-Indentify your asymptotes (where y=0) and draw them on the graph with dotted lines

(Asymptotes for this graph would be: -2π,-π, 0, π and 2π)

- Find the Invariant points again (where sin=1 or sin=-1) in this case they are

-3π/2, -π/2,π/2,3π/2

-Lastly, determine whether sin is +ve or -ve in your invariant points.

If sin is -ve, graph will curve downwards, if it is +ve, graph will curve up

(Use the CAST rule is you have trouble remembering)

YOUR FINAL GRAPH SHOULD LOOK LIKE THIS!

That's it!.....hope these notes help.

oh and tomorrow's blogger will be NM

(cause he missed his turn)

More from today's lessons...March 16th, 2011

Transformations Lesson 4

Drawing Reciprocal Trigonometric Functions Lesson 4b

Secant Graph

Cosecant Graph

Useful Transformations Tutorials...

Secant Graphs Tutorial

Cosecant Graphs Tutorial

Summary for Lesson 10 Graphing Reciprocal Functions

Didn't understand this lesson to the fullest hoping my classmates will explain it better.

An example of a graph of a reciprocal function looks like this http://wpcontent.answcdn.com/wikipedia/commons/thumb/2/29/Rectangular_hyperbola.svg/330px-Rectangular_hyperbola.svg.png

To find the reciprocal of a graph, you would take the original equation ex. y=2x+2 and inverse the "x" side of the equation so it becomes y=1/2x+2

After this set up a table of values for the y and x and the 1/y so that you can plot the original graph points and than its reciprocal function points.

After setting up table of values, solve for variant points by setting value of y to 1 and -1 and solving for x on both of your new equations.

Ex.) y=2x+1

1=2x+1 --> 0=2x --> x=0 y=1 (invariant point)

-1=2x+1 --> -2=2x --> x= -1, y= -1 (invariant point)

The invariant point is to see where the line of the graph will start to turn so it's acting as a turning point for the line.

After plotting invariant points and setting up table of values all thats left to do is plot the reciprocal graph

Make sure to label horizontal and vertical asymptotes and draw arrowheads for both lines, because as a function the line continues outside of the graph.

Transformations Lessons

Here are the last few lessons from our transformations unit.  Enjoy!

Transformations Lesson 2

Transformations Lesson 3

Symmetry, Reflections and Inverses - March 14, 2011

Review:

A graph is a function if all the y values have a unique x value.

If it passes the "Vertical Line Test" it means it is a function.

And a function is said to be a one-to-one function if the inverse of the graph is still a function.

Reflection of Functions:

y=f(x) is what you call your "parent graph". Points and equations will be plotted on the Cartesian plane normally.

If a negative one (-1) is added in front of the "parent" equation, so it changes from y=f(x) to y=-f(x), this means that you will have to reflect the graph over the x-axis. This means that the new points of the graph are equal to the points of the "parent" graph when you fold the plane horizontally at the x-axis.

y = f(x) >>>>>>>>>>>>>>>> y = -f(x)

x = 0 3 3 >>>>>>>>>>>>>>>> x = 0 3 -3

y = 2 3 -1 >>>>>>>>>>>>>>>> y= -2 -3 1

If a negative one (-1) is added in front of the x in your parent graph, meaning your equation changed from y=f(x) into y=f(-x), you have to reflect your graph over the y-axis. So this means that your new points would equal you parent graph's points if you were to fold the whole Cartesian plane vertically on the y-axis.

y = f(x) >>>>>>>>>>>>>>>> y = f(-x)

x = 0 3 3 >>>>>>>>>>>>>>>> x = 0 -3 3

y = 2 3 -1 >>>>>>>>>>>>>>>> y= 2 3 -1

Finally if you try to inverse functions that means your "parent" equation changed into y=f^-1(x) you just interchange your x and y values. Your new points should also equal to the parent graph's new points when you fold the Cartesian plane diagonally on y=x .

y = f(x) >>>>>>>>>>>>>>>> y = f^-1(x)

x = 0 3 3 >>>>>>>>>>>>>>>> x = 2 3 -1

y = 2 3 -1 >>>>>>>>>>>>>>>> y= 0 3 3

Finding the Inverse Equation

To find the inverse equation you need to switch the spots of the y and x in the equation.

If the equation is y=4x-7 you change it into x=4y-7 and then you have to rearrange the new equation isolating y. In this case your new equation will be y=(x+7)/3

If there is a specific value for the inverse, such as f^-1(2) , just find the inverse equation first and then substitute the 2 for the x value.

To find f^-1(2), given f(x)=2x+3

f(x)=2x+3

y=2x+3 >>>>>> y=(x+3)/2

f^-1(x)=(x+3)/2 f^-1(x) = y

f^-1(2)=(x+3)/2 >>>>>>> y=(2+3)/2

y=5/2

Therefore your f^-1(2) , in the equation f(x)=2x+3, is ( 2 , 5/2 )

That's it ! The next blogger will be ......... Nick M.

How to determine the equation of a trigonometric graph

Check out this great video for help with determining the equation of a sine or cosine function given the graph.

 

http://www.youtube.com/watch?v=yvW5l9W1hgE&feature=player_embedded - at=158

Transformations of Sine and Cosine Graphs

At the end of last week we looked at transformations of Sinusoidal graphs.  We looked at how the parameters A, B, C and D affect the position and shape of the graph.

Parameters A B C and D and what they do...

y = A sin f [B(x-C)] + D

A- Vertical Stretch/ Compression
B- Horizontal Stretch/Compression
C- Horizontal Translation
D- Vertical Translation

To find the period of the graph use P=2pi/B.

If B is attached to the x or theta value, you must factor it out of both terms inside the bracket.

For all you visual learners....Check out this great website applet to help you SEE how the parameters change the shape of the graph.  Enjoy.     http://www.ronblond.com/M12/sc.APPLET/index.html

We will continue with lessons on transformations of Cubic functions, Square Root functions and Rational functions.

See you Monday.  Monday's blogger will be.....

"Moving on" with Transformations

Today we had our first unit assessment on Circular Functions.  We will continue on tomorrow with our transformations unit.  We will be looking specifically at Trigonometric functions and their respective graphs.  We will look at how the A, B,C and D in a sinusoidal equation, affect the position and shape of the graph.  We will also look at drawing the inverses of each of these graphs.

Transformations

Today we started our second unit in Grade 12 PreCalculus on Transformations.
We looked at shifting a graph to the left or right, up or down.  When there is a horizontal shift we look for the "h" value.  The shift is opposite to the sign that is in the equation.  When there is a vertical shift we look for the "k" value.  The shift is the same as the sign that is in the equation, plus = up, minus = down.

Transformations Lesson 1 

REMINDER:  CIRCULAR FUNCTIONS UNIT ASSESSMENT  Friday March 5th, 2011 
Date changed to Tuesday March 8,2011