Trig+Functions+Lesson+Summaries


 * Introduction to Radians**

**October 25, 2010**
__Radians__ Radians- A different way of measuring an angle. An angle subtended at the centre of a circle by an arc equal in lenth to the radius. This measurement is in relation to pi (as seen in a unit circle).

In general # of radians = arc length/ radius (x=a/r)

Unit Circle [] Examples

1. How many radians are subtended in a circle of radius seven and arc length of 16?

ϴ=16/7

2. c=2πr ϴ=a/r ϴ=2πr/r ϴ=2π π=180**°** __Converting__ Sometimes it is helpful to convert back and forth from radians and degrees.

Convert to radians: 1a. π/180 = x/30 30**°** = π/6 1b. π/180 = x/45 45**°** = π/4 1c. π/180 = x/127 127π/180 = ? = 2.22 (the unit of the radian is the absense of any unit of measurement)

Convert to degrees: 1a. π/3 = x 180/π = x/π/3 60π/π = x 60**°** = x 1b. 1.73 180/π = x/1.73 99.12**°** = x

__Using Radians in Trig__ Just like degrees we can use radians to calculate angles, sides, etc. Special triangles are often used to express these angles.

Special Triangles [] Ex) Find sin(π/3) or sin(60**°**) = √3/2 or 0.86

__Homework__

Pg 78 # 1-9

__Websites__ For more information on special triangles visit [] For more information on radians visit []

October 26: Graphing Trigonometric Functions !
updated by Mimi Chen

__The graphs of the primary trigonometric functions (in radians) are:__

y = sin x Retrieved from []

y = cos x Retrieved from []

y = tan x Retrieved from []

__Properties of the graphs of sinx, cosx, tanx:__ amplitude: half the difference between the maximum value and minimum value of the function period: the horizontal length of one cycle
 * || maximum || minimum || amplitude || period || y-intercept ||
 * < y = sin x ||= 1 ||= -1 ||= 1 ||= 2π ||= 0 ||
 * < y = cos x ||= 1 ||= -1 ||= 1 ||= 2π ||= 1 ||
 * < y = tan x ||= ∞ ||= -∞ ||= none ||= π ||= 0 ||

__Why does tanx have asymptotes?__
 * tanx is undefined at [[image:http://sphotos.ak.fbcdn.net/hphotos-ak-snc4/hs903.snc4/71654_10150289402880276_821125275_15350223_1826349_n.jpg width="29" height="29"]]where //k// is an odd integer, note k≠ 0
 * since [[image:http://sphotos.ak.fbcdn.net/hphotos-ak-snc4/hs774.snc4/67428_10150289402840276_821125275_15350221_6339486_n.jpg width="81" height="41"]], in order for tanx to be undefined, cosx = 0


 * meaning tan is undefined at [[image:http://sphotos.ak.fbcdn.net/hphotos-ak-ash2/hs412.ash2/69025_10150289402860276_821125275_15350222_4758183_n.jpg width="127" height="40"]]

__Reciprocal Trigonometrical Functions:__

The secondary trigonometrical functions, followed by the reciprocal trig. ratios are depicted below.
 * reciprocal trig. functions are:
 * y = csc x
 * y = sec x
 * y = cot x
 * Reciprocal trig. functions are **__NOT__** the same as inverse trig. functions

For review of reciprocal trigonometrical ratios please check out [] For more information about graphing trigonometrical functions please check out [] []

**Homework:**
- Graph the reciprocal trig. functions (to be checked in class tomorrow!)

updated by: Mariam Naguib
 * October 27th: Even and Odd Trigonometric Functions and TRANSFORMING Trigonometric Functions**


 * __Even or Odd Trigonometric Function__s**

__Recall:__ an //odd function// has double symmetry with the point of symmetry on the origin and an //even function// has a line of symmetry on the y axis making it symmetrical.

Applying that knowledge to Trigonometric Functions To view these functions: [|Trigonometric Functions (MORE INFO!)]
 * Function || Even/ Odd ||
 * sinx || odd ||
 * cosx || even ||
 * tanx || odd ||
 * cscx || odd ||
 * secx || even ||
 * cotx || odd ||

__**Transforming Trigonometric Functions**__

f(x) = a sin [ k ( x - d ) ] + c
a --> tells you if the function is reflected in the x axis, if there is a vertical compression or stretch as well as the amplitude (ie. highest point - lowest point divided by 2 = amplitude)

K --> tells you the period (ie. period = 2π/ k)

-d --> in the opposite direction of 'd's sign a __phase shift__ (aka horizontal translation on a trigonometric function) (therefore go right if negative and left if positive)

c --> tells you that there is a vertical translation (up if positive and down if negative, therefore following c's sign)

﻿Example: Apply to sinx, when the amplitude is 4, the period is π, there is a phase shift of π/6 to the left and a vertical translation of 2 up.

-amplitude of 4, therefore a =4 -period of π, (period (T) = 2π/k, therefore k = 2π/T = 2π/π = 2) therefore k = 2 - phase shift of π/6 to the left, therefore d= + π/6 - vertical translation of 2 up, therefore c = +2

therefore the equation is: f(x) = 4 sin [ 2 (x + π/6) ] +2

For review of even and odd functions check out: [|Even and Odd Functions] For more information of transforming trigonometric functions: [|Tranforming Trigonometric Functions]

Homework: Complete the two photocopies given in class and a Quiz on Tuesday November 2, 2010, so study !

**October 29th: Field Trip! (Gairdner Foundation Student Symposium)**

**November 1st: More Trig!** By: Clinton D'Silva



NOTES FROM CLASS

Any addition help in TRIG? [|Basic Trigonometry] []

Special triangles Video media type="youtube" key="DIGoK51u0KQ?fs=1" height="385" width="480"

QUIZ TOMORROW on Radians and Transformations of trig functions

By: Joanne Kim
 * November 2, 2010: Trig quiz!**

We had a quiz today, and the homework was: pg 83-84 in the math textbook #6-12, and the handout called "Equivalent Trig Functions".

Help with the handout can be found on page 85 in the textbook; the section labeled 4.3 Equivalent Trigonometric Expressions!

By: Travis Guy
 * November 3, 2010: Equivalent Trig Functions**

Class note:

__**Some Helpful Things to Remember**__

- A right angle triangle can be used to find the equivalent trigonometric expression that build up cofunction identities. ie. sin x = cos( pi/2 - x)

- If you are given a trigonometric expression of a known angle, you can the equivalent trig expression to find the trig expressions of other angles.

For a Table of the Trig Identities Featuring pi/2 (90 degrees) See section 4.3 page 85


 * __Friday, November 5, 2010__**
 * Notes by: Aleena Dipede**

__Addition Identities (also known as Compound Angle Formulae)__ Example: Test the compound angle theorem sin(x+y) = sinxcosy + cosxsiny for x = Π/2 and y= 3Π/4



Example: use the addition/subtraction identities to find: a) b)

A useful website to look at... []

Homework: Handout

=Monday, November 8 - Drew=

Proving Trigonometric Identities - Guidelines
1. Treat the left and right sides **SEPARATELY**. So **DO NOT** move things across the equal sign. 2. You will probably need to **FACTOR!** (Things like common factors, differences of squares, trinomials). 3. Unless the left and right sides ONLY deal with tan or cot, change the occurrence of tan or cot into an equivalent trigonometric ratio that is in terms of sin or cos. 4. Change the reciprocal trigonometric ratios (csc, sec, cot) into the **PRIMARY** trigonometric ratios: **sin, cos, and tan**. 5. Get rid of double angles or addition and subtraction of angles using known identities (addition and subtraction of angles is discussed in Aleena's note for November 5 and double angles will be discussed below). 6. __**USE**__ the known identities! 7. Replace sin(-x), cos(-x), -sin(x), and -cos(x) **as appropriate**.

Double Angle Formulae
Ex. sin(2x) OR cos(2x)

sin(2x) = sin(x+x) = sin(x)cos(x)+sin(x)cos(x) [Using the addition identity from the previous day]
 * Derive** sin(2x)
 * = 2sin(x)cos(x)**

cos(2x) = cos(x+x) = cos(x)cos(x)-sin(x)sin(x) = (1-[sin(x)]^2) - [sin(x)]^2
 * Derive** cos(2x)
 * = [cos(x)]^2 - [sin(x)]^2**
 * = 1-2[sin(x)]^2**

OR (from step 4 above) = [cos(x)]^2-(1-[cos(x)]^2) **= 2[cos(x)]^2-1**

[sin(2x)]/(1-cos(2x) [2sin(x)cos(x)]/[1-(1-2[sin(x)]^2)] [2sin(x)cos(x)]/(2[sin(x)]^2) [cos(x)]/[sin(x)] cot(x)
 * Simplify** [sin(2x)]/(1-cos(2x)

OR (from step 2 above) = [2sin(x)cos(x)]/[1-(2[cos(x)]^2-1)] = [2sin(x)cos(x)]/[2-2[cos(x)]^2] = [2sin(x)cos(x)]/[2(1-[cos(x)]^2)] = [2sin(x)cos(x)]/[2[sin(x)]^2] = [cos(x)]/[sin(x)] = cot(x)

Ex) Expand cos(3x) cos(3x) = cos(x+2x) = cos(x)cos(2x) - sin(x)sin(2x) = cos(x)cos(x+x) - sin(x)sin(x+x) = cos(x)([cos(x)]^2-[sin(x)]^2) - sin(x)[2sin(x)cos(x)] = [cos(x)]^3 - [sin(x)]^2cos(x) - 2[sin(x)]^2cos(x) = [cos(x)]^3 - 3[sin(x)]^2[cos(x)] = cos(x){[cos(x)]^2-3[sin(x)]^2}
 * Can't really go any further here, this is just a demonstration that you can mess around with the identities given even without an equal sign.

Ex) Expand sin(4x) sin(4x) = sin(2x+2x) = sin(2x)cos(2x)+sin(2x)cos(2x) = 2sin(2x)cos(2x)

- We received a homework sheet (multiple choice questions like the addition and subtraction identities sheet). - We also received a handout with the various trigonometric identities that we learned. **Gillian said that it would probably be photocopied and put onto our unit test!** - Tuesday we are having a work period (for solving trigonometric identities) and receiving an assignment. **PRACTICE** them! - There is a **QUIZ** on Wednesday on solving trigonometric identities (3 questions).
 * Handouts/Homework:**


 * Links:**
 * [|Trigonometric Identities Table]**
 * [|Solving Trigonometric Identities]**
 * [|Goldmine of Trigonometry!]**

=November 9 - Trig. Identity Examples= by Mimi Chen



Also, today was Teamwork Tuesday! We did an activity where we had a square sheet of paper, we cut the largest possible circle out of the square paper, and then cut the largest possible square out of that circle. We were to determine what fraction of the original square was the final square. which turned out to be one half. The rest of the period was a work period.

For more trig. identity help please check out [] and [] Also, check out [] for more advice and a few examples! Check out [|for sum and difference of angles identities] and [|for double angle identities]

__**Homework:**__ Verify the following Trigonometric Identities worksheet Trigonometric Identities Assignment (due Friday)

Quiz Tomorrow!!!!!
=November 10 2010 - Solving Trig Equations= =Updated by Sarah Drury=

Solve, if 0 ≤ x ≤ 2π a) sinx = ½ x = π/6, 5π/6  b) 2 cosx + 1 = 0 2 cosx = -1 cos x = -½ x = 2π/3, c) 3(tanx+1) = 2 tanx +1 = 2/3  tanx = -1/3  x = tan -1( -1/3)  x = 2π - 0.322 or x = π - 0.322  x = 5.96, 2.818

d) sin 2 x - 1 = 0

sin 2 x = 1 sin x = ± 1 x = sin -1 (± 1) x = π/2, 3π/2 e) sin 2 x - sinx = 2   sin 2 x - sinx - 2 = 0   (sinx-2)(sin+1) = 0  sinx= 2 or sinx= -1  *Can't be 2 because sinx never reaches 2  sinx= -1  - Use special triangles when you can  - Remember the CAST rule  - Watch out for restrictions - Draw the angle/graph to make things easier
 * __Things to remember__: **

Page 109
 * Homework:**
 * 2,6

For more help: [|Cool Math Trig Lesson] [|Int Math Solving Trig Equations Lesson] [|Purple Math Solving Trig Equations Lesson]

**Topic**: __Solving MORE Complicated Trigonometric Equations__ **Date**: November 12th 2010 **Name**: Joanna Decc

Solve for: 0 ≤ x ≤ 2π a) 3sinx = sinx + 1 2sinx = 1 sinx = 1/2 x = π/6, 5π/6

b) cos2x = -1/2 2x = cos^-1 (-1/2) 2x = 2π/3 OR x = 4π/3 x = 2π/6 OR x = 4π/6 x = π/3 OR x = 2π/3 + π = 5π/3 + π/4π/3

Solve for 0 ≤ x ≤ 2π c) 3sinx + 3cos2x = 2 3sinx + 3(1-2sin^2x) = 2 sinx + 1-2sin^2x = 2/3 -2sin^2x + sinx + 1/3 = 0 6sin^2x - 3sinx - 1 = 0 sinx = 0.72 OR -0.23 x = 0.8, 2.34 OR x = -0.23, 6.05, 3.37


 * Homework**: Worksheet- //Exercises 9.4// (circled questions only!)

Tomorrow is our Unit Test in Trigonometry! :) Gillian handed out a practice quiz and practice for our trig identities. I have attached a unit review, remember guys we're at the top of the mountain! //Good luck Studying//! I hope this review is helpful-- Keep it for exam time! :)
 * __Monday November 15__**


 * UNIT TEST**: __Tuesday November 16th__

Need more Help?
 * @http://oakroadsystems.com/math/trigsol.htm
 * @http://mathscene.co.uk/t/F/Su36k05.htm
 * @http://www.mathslideshow.com/Alg2/Lesson14-4/index.htm