FURTHER MATHS rotation matrices help

hey im trying to teach myself OCR MEI's FM1 (gap year) and have only just started (not planning on taking the exam, just learning the content)
im trying to understand how to find the matrix which represents a rotation through angle (theta) anticlockwise about the origin.

im using the mei FM1 third edition textbook trying to understand activity 1.3 in chapter 1

i read through and sort of understood the example about using points (0,1) and (1, 0) as example points to rotate where you make a right angled triangle and use trig to find x and y values of the new points in terms of sin and cos theta. or at least i thought i did.

then the textbook asked me to 'investigate rotation matrices for angles between 90 and 180 degrees, angles between 180 and 270 degrees, etc' and now im really confused.
i know that basically the answer is going to mainly be whether the cos(theta)s and sin(theta)s will be positive or negative, but i just cant even begin to understand how to work it out.
i wish i could write more about what my problem is but i dont even understand it enough to work out what bit i dont understand

like, how do you decide whether a or b should represent the height or width of the right angled triangle?

is a always equal to costheta and b always sintheta?

also the answers are just ' costheta is positive, sintheta is negative' or something ? how is that in any way an answer to the question 'investigate rotation matrices for angles between 90 and 180 degrees'?

im just so confused. should i give up on trying to teach myself further maths if im so confused about this stuff already?

used to think i was good at maths but now i just feel like a useless idiot :-)

Reply 1

joostan

13

Original post by EmergencyBagels

hey im trying to teach myself OCR MEI's FM1 (gap year) and have only just started (not planning on taking the exam, just learning the content)
im trying to understand how to find the matrix which represents a rotation through angle (theta) anticlockwise about the origin.

im using the mei FM1 third edition textbook trying to understand activity 1.3 in chapter 1

i read through and sort of understood the example about using points (0,1) and (1, 0) as example points to rotate where you make a right angled triangle and use trig to find x and y values of the new points in terms of sin and cos theta. or at least i thought i did.

then the textbook asked me to 'investigate rotation matrices for angles between 90 and 180 degrees, angles between 180 and 270 degrees, etc' and now im really confused.
i know that basically the answer is going to mainly be whether the cos(theta)s and sin(theta)s will be positive or negative, but i just cant even begin to understand how to work it out.
i wish i could write more about what my problem is but i dont even understand it enough to work out what bit i dont understand

like, how do you decide whether a or b should represent the height or width of the right angled triangle?

is a always equal to costheta and b always sintheta?

also the answers are just ' costheta is positive, sintheta is negative' or something ? how is that in any way an answer to the question 'investigate rotation matrices for angles between 90 and 180 degrees'?

im just so confused. should i give up on trying to teach myself further maths if im so confused about this stuff already?

used to think i was good at maths but now i just feel like a useless idiot :-)

It's tricky to explain without a diagram, I may draw one later if this doesn't help.

Draw yourself some axes, and an additional set which you rotate to, and mark the angle

\theta

as done for the x-axis in the link.
Hopefully you can see when to use sine and cosine from the diagram.

(edited 8 years ago)

Reply 2

Zacken

22

Original post by EmergencyBagels

...

I'm not sure I understand? You've derived the fact that matrix representation of a rotation of

\theta

(anticlockwise) about the origin is given by:

Unparseable latex formula:

\displaystyle[br]\begin{equation*}\begin{pmatrix}\cos \theta & -\sin \theta \\ \sin \theta & [br]\cos \theta \end{pmatrix}\end{equation*}

haven't you? That works for any

\theta \in [0, 2\pi]

.

Reply 3

EmergencyBagels

OP

18

Original post by Zacken

I'm not sure I understand? You've derived the fact that matrix representation of a rotation of

\theta

(anticlockwise) about the origin is given by:

Unparseable latex formula:

\displaystyle[br]\begin{equation*}\begin{pmatrix}\cos \theta & -\sin \theta \\ \sin \theta & [br]\cos \theta \end{pmatrix}\end{equation*}

haven't you? That works for any

\theta \in [0, 2\pi]

.

lol i dont understand me either man

but soo.. yeah i think the book example says that.. i guess i just dont really get what that means or what thats meant to prove?

oh man i just kind of got what it was asking me now! ok i need some time to let my brain recover from this and look back over it but i think i get it now! if not then i suppose i will have to ask some more confusing questions in the near future hah

Reply 4

LelouchViRuge

7

Original post by EmergencyBagels

lol i dont understand me either man

but soo.. yeah i think the book example says that.. i guess i just dont really get what that means or what thats meant to prove?

oh man i just kind of got what it was asking me now! ok i need some time to let my brain recover from this and look back over it but i think i get it now! if not then i suppose i will have to ask some more confusing questions in the near future hah