VCE Specialist 3/4 Question Thread!

[eagles]

Hi there!

I'm having trouble with question 11b) (see attached). Can someone please help?

Also I'm not sure how to visualise this question:

A curve on a light track is an arc of a circle of length 300m and the straight line joining the two ends of the curve is 270m long. Show that, if the arc subtends an angle of 2θ^o at the centre of these circle, θ is a solution of the equation sin θ⁰ = pi/200 θ^o.
Solve, correct to 2 decimal places, the equation for θ.

Cheers!

[lzxnl]

Hi there!

I'm having trouble with question 11b) (see attached). Can someone please help?

Also I'm not sure how to visualise this question:

A curve on a light track is an arc of a circle of length 300m and the straight line joining the two ends of the curve is 270m long. Show that, if the arc subtends an angle of 2θ^o at the centre of these circle, θ is a solution of the equation sin θ⁰ = pi/200 θ^o.
Solve, correct to 2 decimal places, the equation for θ.

Cheers!

cot x = cos(x)/sin(x)=sin(pi/2-x)/cos(pi/2-x)=tan(pi/2-x)
See if that helps

As for your second question...draw an isosceles triangle with side lengths 300m (for two of them) and 270m. Then, draw a perpendicular bisector from the 270m side to the opposite corner. Try the rest now.

[eagles]

Yep! Thanks for the hints xD

I was also wondering whether sin^-1(sin(5pi/4)) can be evaluated?
If not do we write "solution is undefined"?

Thank you

[lzxnl]

Yep! Thanks for the hints xD

I was also wondering whether sin^-1(sin(5pi/4)) can be evaluated?
If not do we write "solution is undefined"?

Thank you

well...sin(5pi/4)=-sin(pi/4)=sin(-pi/4). Do you think the inverse sine of this is undefined?

[eagles]

Oh gee... haha nice one 

[brightsky]

in general arcsin(sinx) is defined for x E R, but arcsin(sinx) = x only if x is within the domain [-pi/2, pi/2].

[eagles]

In general, how do we find the Argument of complex numbers?

For example, how would we determine the Argument for -8+15i and -4-3i?

My interpretation is that we draw a triangle in whichever quadrant it lies and then find the acute angle in the triangle. For -8+15i, my answer is -tan^-1(15/8) = 3.18 while the answer has 2.06.
For -4-3i, my answer is tan^-1(3/4) = 0.64 while the answer is -2.50.

I am not sure where I went wrong so I would really appreciate some help! Thanks!

[Eugenet17]

For -8+15i, it would be in the 2nd quadrant with coordinates (-8,15) on the Argand diagram, you can draw this line and make a triangle as you said. I would find the 1st Q angle (tan^-1(15/8)=1.08) and just do pi-1.08 as its in the 2nd quadrant which equals 2.06.

[eagles]

Ahh.. I got it now thanks!
So for the second example, -4-3i, my answer is tan^-1(3/4) = 0.64 and I subtract pi to get -2.50.

[eagles]

How would we show that |z| = 1?

I've attached the worked solutions but having trouble understanding the steps.
Where did the 'i' go?

Thanks!

[Eugenet17]

When you're finding |z| or the modulus of z, the formula is sqrt(Re(z)^2+Im(z)^2), you're only squaring the real and imaginary components ( i.e. Re(z) = (costheta) and Im(z) = sin(theta) or in simpler terms for instance z= 1+2i, Re(z)=1, Im(z)=2)

From there you get the trig identity cos^2x+sin^2x=1

[eagles]

Thanks so much!

[Eugenet17]

how do I differentiate sec^2(5x^2)? :x

[RKTR]

how do I differentiate sec^2(5x^2)? :x

1/cos^2(5x^2). Use quotient rule

[Eugenet17]

This is how I ended up doing it, can someone tell me if I would get full marks for working out, or do I have to state to Let v=cos(u) , u=5x^2 and do dy/dx=dy/dv. dv/du . du/dx?

Using Chain Rule: