try or get her turf? i’m on!! there r torn fruit!? o my g!!! try or get her turf? i’m on!! there...
TRANSCRIPT
Try or get her turf? I’m on!! Try or get her turf? I’m on!! There r torn fruit!? O my G!!!There r torn fruit!? O my G!!!
Try or get her turf? I’m on!! Try or get her turf? I’m on!! There r torn fruit!? O my G!!!There r torn fruit!? O my G!!!
By By DUKE DUKE LAME axe York LAME axe York
My real axe duke ok?My real axe duke ok?
Trigonometry review
hyp
oppsin
Hypotenuse
Opposite
Adjacenthyp
adjcos
adj
opptan
Trigonometric Identities
1cossin 22 )90(cossin
cos
sintan
)90(sincos cossin22sin
22 sincos2cos
Aim of this chapter• Definition of cos x and sin x in terms of
the unit circle• How to prove all the identities• The circular functions sin x and cos x
and tan x, their domains, ranges, their periodic nature, and their graphs
• Solution of trigonometric equations in a finite interval
Looking at y=sin x
Properties of y=sin x• f(x)=sin x is periodic of 2π• f(x)=sin x has rotational symmetry
about the origin of order 2• The range of f(x)= sin x is
1)(1 xf
Looking at y=cos x
Properties of y=cos x• The function f(x) = cos x is
periodic, of period 2π rad.• The graph of f(x) = cos x is
symmetrical about the y axis• The range of the function f(x) =
cos x is between 1 and -1
1)(1 xf
Looking at y = tan x
Properties of the tangent function
Y=tan x (radians)• The function f(x) = tan x is periodic, of
period π rad.- tan (x+ π) = tan x
• The graph has a rotational symmetry about the origin of order 2
• The function f(x) = tan x is no defined when
....2
3,
2
x
Perfect makes Practice
• Exercise 8A on Page 238• Questions 1-4, do at least 3
questions on each section.• Use the examples to help you
ONLY IF YOU REALLY NEED TO.
ASTC – Trigonometric Equations
• Trigonometric equations may have more than 1 answer.
• Plot y=cos x and y=1/2• Look between the ranges• How many values of x are there?
22 x
ASTC - Continued
AS
T C
Perfect makes Practice• Exercise 8B on Page 247• Questions 1,3,4,6 do at least 3
questions on each section if there are more than 3.
• Use the examples to help you ONLY IF YOU REALLY NEED TO.
Important things to remember when solving more complicated trig
equations.• Change the range!! If the equation
asks for sin 2x, change the range to 2x.
• Sometimes you may need to FACTORISE.
Prefect makes Practice• Exercise 8C on Page 251• Questions 1-3 do at least 3
questions on each section.• Use the examples to help you
ONLY IF YOU REALLY NEED TO.
Proving Trigonometric identities
Hint: Draw a right angle triangle with sides a,b,c.
)90(cossin
)90(sincos
Proving more trigonometric identities•
• Draw a right angle triangle with sides a, b, c
• Write the equation for sin θ and cos θ.
• Prove it!!!
cos
sintan
Proving even more trigonometric identities• • Draw a right angled triangle with sides
a, b, c• Write the equation for sin θ and cos θ.• Maybe we should square it and see
what happens…• Remember Pythagoras?• Prove it!!!!!
1cossin 22
Perfect practice makes Perfect
• Exercise 8D on Page 254• Questions 1,6,7,8,9 do at least 3
questions on each section if there are more than 3.
• Use the examples to help you ONLY IF YOU REALLY NEED TO.
Double angles
Key points!!! (remember them)• Sin 2x = 2sin x cos x• Cos 2x = cos2x - sin2x
Feeling the urge to prove those? XDI know you do ..
c
xxa
b
de
First, find in terms of x, what sin 2x = ?
Secondly, find in terms of x, what Cos 2x = ?
Now, substitute x in and see if you are right!!
If you do not trust your own proof
You can check your answer with your GDC!!
• Plot the graph (y=sin2x), and (y = 2sinx cosx) What do you notice?Clear your graphs and now plot the graphs• Y = Cos 2x , y=cos2x - sin2x• y = 2cos2x – 1• y = 1 – 2sin2x
If you can’t be bothered to do so, just trust your own proof XD
Prefect makes perfect practice
• Exercise 8E on Page 258• Questions 1,3,5,8 do at least 3
questions.• Then finish 9-15 for more
challenging practices.• Use the examples to help you ONLY IF YOU REALLY NEED TO.
Monkey spotMonkeys have learnt their lesson
Summary
The golden question
The hardest question in the world
What was the title on slide 20?
a) Perfect makes practiceb) Perfect practice makes perfectc) Prefect makes perfect practiced) Prefect makes practicee) Practice makes prefect perfectf) What was the title on slide 20?g) Proving trigonometric identities.h) Proving even more trigonometric
identitiesi) Double anglesj) The end
Zoo
Pencil case
The End
• Cos 2x = 2cos2x – 1• Cos 2x = 1 – 2sin2x