My Math Forum  

Go Back   My Math Forum > High School Math Forum > Trigonometry

Trigonometry Trigonometry Math Forum


Thanks Tree3Thanks
  • 1 Post By v8archie
  • 1 Post By skipjack
  • 1 Post By skeeter
Reply
 
LinkBack Thread Tools Display Modes
December 11th, 2018, 05:57 AM   #1
Newbie
 
Joined: Dec 2018
From: Canada

Posts: 14
Thanks: 0

Math Focus: mainly algebra
Factoriser, spent 5 hours with it

i understand this might be easy even if I spent a lot of time to get a clue, but there is a beginning to everything so here it is if anyone can help me:

Factorise:
1-sin(x).cos(2x)

Thanks.

Last edited by skipjack; December 11th, 2018 at 07:59 AM.
SkyCod is offline  
 
December 11th, 2018, 06:15 AM   #2
Math Team
 
Joined: Dec 2013
From: Colombia

Posts: 7,681
Thanks: 2659

Math Focus: Mainly analysis and algebra
$$\cos 2A = \cos^2 A - \sin^2 A = 1 - 2\sin^2 A$$
This leads to a cubic in $\sin x$ that can be factorised.
Thanks from SkyCod
v8archie is offline  
December 11th, 2018, 07:17 AM   #3
Newbie
 
Joined: Dec 2018
From: Canada

Posts: 14
Thanks: 0

Math Focus: mainly algebra
Yes, I did follow that and with no actual lead if you put it on paper, you can try it.

Last edited by skipjack; December 11th, 2018 at 07:58 AM.
SkyCod is offline  
December 11th, 2018, 07:52 AM   #4
Global Moderator
 
Joined: Dec 2006

Posts: 20,968
Thanks: 2217

The above method leads to (sin(x) + 1)(2sin²(x) - 2sin(x) + 1).
Thanks from SkyCod
skipjack is offline  
December 11th, 2018, 11:42 AM   #5
Newbie
 
Joined: Dec 2018
From: Canada

Posts: 14
Thanks: 0

Math Focus: mainly algebra
Quote:
Originally Posted by skipjack View Post
The above method leads to (sin(x) + 1)(2sin²(x) - 2sin(x) + 1).
How come it does? And even if it did, it's still not a complete factorisation.

Last edited by skipjack; December 11th, 2018 at 01:24 PM.
SkyCod is offline  
December 11th, 2018, 12:15 PM   #6
Math Team
 
skeeter's Avatar
 
Joined: Jul 2011
From: Texas

Posts: 3,016
Thanks: 1600

Quote:
Originally Posted by SkyCod View Post
How come it does? And even if it did, it's still not a complete factorisation.
$1-\sin{x}\cos(2x)$

$1-\sin{x}(1-2\sin^2{x})$

$1 - \sin{x} + 2\sin^3{x}$

$(1 + \sin^3{x}) - (\sin{x} - \sin^3{x})$

$(1+\sin{x})(1-\sin{x}+\sin^2{x}) - \sin{x}(1-\sin^2{x})$

$\color{red}{(1+\sin{x})}(1-\sin{x}+\sin^2{x}) - \sin{x}(1-\sin{x})\color{red}{(1+\sin{x})}$

$\color{red}{(1+\sin{x})}\bigg[(1-\sin{x}+\sin^2{x}) - \sin{x}(1-\sin{x})\bigg]$

$(1+\sin{x})\bigg[1-\sin{x}+\sin^2{x} - \sin{x}+\sin^2{x}\bigg]$

$(1+\sin{x})(1-2\sin{x}+2\sin^2{x})$

that's it ...
Thanks from topsquark

Last edited by skipjack; December 11th, 2018 at 01:24 PM.
skeeter is offline  
December 11th, 2018, 12:18 PM   #7
Global Moderator
 
Joined: Dec 2006

Posts: 20,968
Thanks: 2217

You asked for help in factorizing, not for a complete factorization, but skeeter's provided more detail for you.
skipjack is offline  
December 11th, 2018, 12:45 PM   #8
Math Team
 
Joined: Dec 2013
From: Colombia

Posts: 7,681
Thanks: 2659

Math Focus: Mainly analysis and algebra
Quote:
Originally Posted by SkyCod View Post
How come it does? And even if it did, it's still not a complete factorisation.
It is a complete factorisation with real coefficients. The polynomial $2x^2-2x+1$ is irreducible because $b^2 - 4ac = 4 - 8 = -4 <0$

Last edited by skipjack; December 11th, 2018 at 01:23 PM.
v8archie is offline  
December 11th, 2018, 01:22 PM   #9
Global Moderator
 
Joined: Dec 2006

Posts: 20,968
Thanks: 2217

Quote:
Originally Posted by SkyCod View Post
How come it does?
$\displaystyle \begin{align*}2\sin^3(x) - \sin(x) + 1 &= 2\sin(x)(\sin^2(x) - 1) + \sin(x)+ 1 \\
&= 2\sin(x)(\sin(x) + 1)(\sin(x)- 1) + \sin(x) + 1 \\
&= (\sin(x) + 1)(2\sin(x)(\sin(x) - 1) + 1) \\
&= (\sin(x) + 1)(2\sin^2(x) - 2\sin(x) + 1) \\
&= (\sin(x) + 1)((1+ i)\sin(x) - 1)((1 - i)\sin(x) - 1)\end{align*}$
skipjack is offline  
Reply

  My Math Forum > High School Math Forum > Trigonometry

Tags
factoriser, hours, spent



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
I'm out after 2 hours for this question helen510 Geometry 7 April 3rd, 2018 03:09 AM
how much should be spent on each to optimize output? puppypower123 Calculus 1 March 27th, 2017 03:51 PM
How many hours does it take to make.. Pumaftw Elementary Math 2 October 6th, 2014 09:17 AM
Expand and simplify (I've spent 5 hours trying to solve this sharp Algebra 9 December 21st, 2010 03:39 PM





Copyright © 2019 My Math Forum. All rights reserved.