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)
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.
- Graph the reciprocal trig. functions (to be checked in class tomorrow!)
October 27th: Even and Odd Trigonometric Functions and TRANSFORMING Trigonometric Functions
updated by: Mariam Naguib
Even or Odd Trigonometric Functions
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
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
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)
QUIZ TOMORROW on Radians and Transformations of trig functions
November 2, 2010: Trig quiz!
By: Joanne Kim
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!
November 3, 2010: Equivalent Trig Functions
By: Travis Guy
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)
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)
Derive sin(2x)
sin(2x)
= sin(x+x)
= sin(x)cos(x)+sin(x)cos(x) [Using the addition identity from the previous day] = 2sin(x)cos(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.
Handouts/Homework:
- 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).
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.
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) sin2x - 1 = 0
sin2x = 1 sin x = ± 1 x = sin -1 (± 1) x = π/2, 3π/2 e) sin2x - sinx = 2 sin2x - sinx - 2 = 0 (sinx-2)(sin+1) = 0 sinx= 2 or sinx= -1 *Can't be 2 because sinx never reaches 2 sinx= -1 Things to remember: - Use special triangles when you can - Remember the CAST rule - Watch out for restrictions
- Draw the angle/graph to make things easier
Monday November 15
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! :)
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
http://picsdigger.com/keyword/trig%20unit%20circle/
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
http://www.gradeamathhelp.com/image-files/special-right-triangles.gif
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 http://www.purplemath.com/modules/trig.htm
For more information on radians visit http://www.themathpage.com/atrig/radian-measure.htm
October 26: Graphing Trigonometric Functions !
updated by Mimi ChenThe graphs of the primary trigonometric functions (in radians) are:
y = sin x
Retrieved from http://www.themathpage.com/atrig/graphs-trig.htm
y = cos x
Retrieved from http://www.themathpage.com/atrig/graphs-trig.htm
y = tan x
Retrieved from http://www.themathpage.com/atrig/graphs-trig.htm
Properties of the graphs of sinx, cosx, tanx:
period: the horizontal length of one cycle
Why does tanx have asymptotes?
Reciprocal Trigonometrical Functions:
The secondary trigonometrical functions, followed by the reciprocal trig. ratios are depicted below.
For review of reciprocal trigonometrical ratios please check out http://www.purplemath.com/modules/basirati.htm
For more information about graphing trigonometrical functions please check out
http://www.intmath.com/Trigonometric-graphs/4_Graphs-tangent-cotangent-secant-cosecant.php
http://www.purplemath.com/modules/triggrph.htm
Homework:
- Graph the reciprocal trig. functions (to be checked in class tomorrow!)October 27th: Even and Odd Trigonometric Functions and TRANSFORMING Trigonometric Functions
updated by: Mariam Naguib
Even or Odd Trigonometric Functions
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
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
http://www.free-ed.net/free-ed/Math/Trigonometry/trig02_SPK.asp
Special triangles Video
QUIZ TOMORROW on Radians and Transformations of trig functions
November 2, 2010: Trig quiz!
By: Joanne Kim
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!
November 3, 2010: Equivalent Trig Functions
By: Travis Guy
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...
http://www.sosmath.com/trig/addform/addform.html
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)Derive sin(2x)
sin(2x)
= sin(x+x)
= sin(x)cos(x)+sin(x)cos(x) [Using the addition identity from the previous day]
= 2sin(x)cos(x)
Derive cos(2x)
cos(2x)
= cos(x+x)
= cos(x)cos(x)-sin(x)sin(x)
= [cos(x)]^2 - [sin(x)]^2
= (1-[sin(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
Simplify [sin(2x)]/(1-cos(2x)
[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)
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)
Handouts/Homework:
- 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).
Links:
Trigonometric Identities Table
Solving Trigonometric Identities
Goldmine of Trigonometry!
November 9 - Trig. Identity Examples
by Mimi ChenAlso, 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 http://www.mathwizz.com/algebra/help/help32.htm and http://www.purplemath.com/modules/idents.htm
Also, check out http://www.intmath.com/analytic-trigonometry/1-trigonometric-identities.php 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) sin2x - 1 = 0
sin2x = 1
sin x = ± 1
x = sin -1 (± 1)
x = π/2, 3π/2
e) sin2x - sinx = 2
sin2x - sinx - 2 = 0
(sinx-2)(sin+1) = 0
sinx= 2 or sinx= -1
*Can't be 2 because sinx never reaches 2
sinx= -1
Things to remember:
- Use special triangles when you can
- Remember the CAST rule
- Watch out for restrictions
- Draw the angle/graph to make things easier
Homework:
Page 109
#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!)
Monday November 15
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! :)
UNIT TEST: Tuesday November 16th
Need more Help?