Try or get her turf? I’m on!! There r torn fruit!? O my G!!! Try or get her turf? I’m on!! There r torn fruit!? O my G!!! By DUKE LAME axe York My real axe duke ok?

Trigonometry review Hypotenuse Opposite Adjacent

Trigonometric Identities

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

Looking at y=cos x

Properties of y=cos x The function f(x) = cos x is periodic, of period 2 π 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

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

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?

ASTC - Continued A S 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.

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!!!

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!!!!!

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 = cos 2 x - sin 2 x

Feeling the urge to prove those? XD I know you do.. c x x a b d e 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 spot Monkeys have learnt their lesson

Summary

The golden question

The hardest question in the world What was the title on slide 20? a) Perfect makes practice b) Perfect practice makes perfect c) Prefect makes perfect practice d) Prefect makes practice e) Practice makes prefect perfect f) What was the title on slide 20? g) Proving trigonometric identities. h) Proving even more trigonometric identities i) Double angles j) The end

Zoo

Pencil case

The End Cos 2x = 2cos 2 x – 1 Cos 2x = 1 – 2sin 2 x