DannyN's SM question thread =D

[DannyN]

hey guys thought it would be a better idea to make a thread since im always stuck

Vectors:
Given that OP=p, OQ=q and the points O,P,Q are not collinear, which one of the following points, whose position vectors are given, is not collinear with P and Q


a)1/2p+1/2q
b)3p-2q
c)p-q
d)1/3p+2/3q
or
e)2p-q

thanks~

[evaever]

c)

[Greatness]

c)

Maybe explain it
p-q=-(q-p) which is the vector from Q to P. Which is therefore not colinear with either P or Q.

[evaever]

c)

Maybe explain it
p-q=-(q-p) which is the vector from Q to P. Which is therefore not colinear with either P or Q.

?

[DannyN]

Thanks guys ^^ ,okay next question :S

Find the modulus of 1+cos(2theta)+isin(2theta) where 0<theta<pi/2

[evaever]

Thanks guys ^^ ,okay next question :S

Find the modulus of 1+cos(2theta)+isin(2theta) where 0<theta<pi/2

2cosx

[DannyN]

can you please explain to me briefly how you did that? it's amazing how you get the answers so quickly ^^

[evaever]

can you please explain to me briefly how you did that? it's amazing how you get the answers so quickly ^^

draw 1+z where z=cis2x

[evaever]

the arg is x

[luffy]

Just thought I would state things slightly more clearly - Below is how I would do this question. (Sorry if I made any errors).







You can then factorise the expression by taking out a .





Recall that in the expression, , That same principle applies to this question, where r = .







Hope I helped.

[brightsky]

Another way:
sqrt((1 + cos(2t))^2 + (sin(2t)^2))
= sqrt(1 + 2cos(2t) + cos^2(2t) + cos^2(t))
= sqrt(2 + 2cos(2t))
= sqrt(4 cos^2(t))
= |2cos(t)|

[DannyN]

Awesome thanks guys!

[DannyN]

another one
if acos^2(theta)+bsin^2(theta)=c then tan^2(theta) in terms of a,b and c is:

[luffy]

My method seems a bit long-winded, but this would be my solution:



First, solve for , therefore:







Do the same for , therefore:







Now, solve   :









Lets hope I made no mathematical errors.

Hope I helped.

[DannyN]

thanks luffy! =D very helpful ^_^

stuck on another...
sin(2cos^-1(2x)) is defined for x an element of (root(2)/4,1/2)
it can be shown that sin(2cos^-1(2x)=ax(square root(1-b^2x^2) where a and b are positive constants
find a and b
sorry about the text i dont know how to write the square root sign