VCE Specialist 3/4 Question Thread!

[Cort]

Just to give myself a placebo effect - would anyone explain to me why, not how, compound angles / double angles is important in the realm of mathematics? The chapter I'm doing now I can understand, but what is the underlying purpose to it? Compare the ratios of angles? Furthermore, what relevance does it have: in real life (where about(s) in Engineering), and, in mathematics? i.e foundation to understanding something later that I'll be encountering later in Spesh?

Lastly, could the knowledge I reap from specialist in the trig be applied in methods as well?

Thanks,
Cort.

[b^3]

The first thing that comes to mind is when you start dealing with waves and standing waves in a mathematical sense in physics. You may have two waves that are interfering with each other that are out of phase, which is where the compound angle formulas can come in to help simplify the situation. Extending this to engineering, you can get forced oscillations problems where they'll come up and simplify the problem, turning a complex combination of sines and cosines into a simple form which is a hell of a lot easier to work with (talking turning 5 pages of work into about half a page).


Really later (talking uni level, which is most of what I'm talking about above) on you won't be using them to compare angles, but rather using them to help simplify situations where products of sine and cosine occur.

For spesh you might use them a bit later to get out of tricky situations in calculus, when you're trying to integrate something that looks awful, but has a much prettier and elegant look when you realise what you can turn it into, which in turn makes the solution a lot easier and nicer.

I guess also the derivation of cis uses them a bit to simplify things.

tl;dr, they're worth it, they get you out of ugly situations later on.

[lzxnl]

The compound angle formulas are essential in proving that the derivative of sin x is cos x. They can also neaten some expressions as b^3 has mentioned. VCE maths doesn't really tell you the uses of all the formulas it teaches you; just accept that in physics particularly, everything you learn in spesh will be useful.

[eagles]

Hi guys I have a silly question:

The points (8,4) and (2,2) are the ends of a diameter of a circle.

Find the coordinates of the centre and the radius of the circle.

Any help is appreciated. Cheers

[alchemy]

...why compound angles / double angles is important in the realm of mathematics? ..... what is the underlying purpose to it? .... what relevance does it have: in real life

                                                         
               

 

[nhmn0301]

Hi guys I have a silly question:

The points (8,4) and (2,2) are the ends of a diameter of a circle.

Find the coordinates of the centre and the radius of the circle.

Any help is appreciated. Cheers

I would use some Methods equations to find the solution for that one:
If you can recall your mid point equation, (  (x1 + x2)/2 , (y2 + y2) /2  )
Hence, the centre is (5, 3)
Then just continue to use Methods equations to find the distance between 2 points:
d = sqrt( (x2-x1)^2 + (y2-y1)^2)
Hence d= sqrt 40 = 2 sqrt 10
r = 2 sqrt 10/2 = sqrt 10
Hope this helps. There might be other ways but I find this is easier for me.

[alchemy]

Hi guys I have a silly question:

The points (8,4) and (2,2) are the ends of a diameter of a circle.

Find the coordinates of the centre and the radius of the circle.

Any help is appreciated. Cheers

Let (8,4) be point P, and (2,2) be point Q. Find the distance between these two points to find the diameter. So sqrt((y₂-y₁)² + (x₂-x₁)²) = sqrt((2-4)²⁺(2-8)²) = sqrt(4+36) = sqrt(40). The radius is half of this, so it would be sqrt(40)/2 = sqrt(10). To find the centre: (8-2)/2=3 and (4-2)/2=1. So the center would be (2+3,2+1) = (5,3).

EDIT: Beaten.

[eagles]

Yep got it now I was incorrectly assuming they were vertices and couldn't quite get the answer.

Thank you.

[T-Infinite]

Hey guys, please help me with this question.

How do you solve this step by step?

, the answer is

[b^3]

Technically you're simplifying the expression rather than solving for something


EDIT: Missed a square under the square root, fixed.

[T-Infinite]

Technically you're simplifying the expression rather than solving for something

Ooo thank you ! yeah I meant simplifying, silly me! hahah

[T-Infinite]

Need a little help for part c and d of this particular question.

a small asteroid has position vector
r(t)= (2t+1)i + (t^2-11)j +(t^2-3)k at time t E R

a space station has position vector
s= i-12j-2k

c) find the closest distance that the asteroid comes to the space station 
d) The missile is to be launched from the space station to hit the asteroid at time t=-1. If the missile is launched at time t=-2, what angle will its path (assumed to be a straight line) need to make with the line from the space station to the asteroid? Give answer to the nearest degree.

My guess is that for c, it involves the perpendicular thingo and for d do we use the formula with cos

[Ancora_Imparo]

Need a little help for part c and d of this particular question.

a small asteroid has position vector
r(t)= (2t+1)i + (t^2-11)j +(t^2-3)k at time t E R

a space station has position vector
s= i-12j-2k

c) find the closest distance that the asteroid comes to the space station

Let the distance between the the asteroid and space station be D.










For minimum distance, :



When :
units

Thus, the minimum distance between the asteroid and space station is units.

[T-Infinite]

Let the distance between the the asteroid and space station be D.










For minimum distance, :



When :
units

Thus, the minimum distance between the asteroid and space station is units.

How come for minimum distance, : ??

[Ancora_Imparo]

How come for minimum distance, : ??

To find the find minimum value of D, we differentiate D with respect to t and equate to 0 to find the minimum turning point. Since the function is a quadratic (), we didn't have to do this - we could have just said by inspection that the minimum would occur at the turning point, which is at t=0.

d) The missile is to be launched from the space station to hit the asteroid at time t=-1. If the missile is launched at time t=-2, what angle will its path (assumed to be a straight line) need to make with the line from the space station to the asteroid? Give answer to the nearest degree.

When :


Path that missile has to take is given by the vector between space station and asteroid when :


When :


Initial path between space station and asteroid is given by vector between space station and asteroid when :


Let the angle between the two vectors be . Using dot product:







You then use your calculator to find .