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trigonometry and complex number questions

Hello everyone, I desperately need help in a paper I solved recently. Here are the two questions I need help in...

In q8, I have problem in finding least and greatest possible values in ii and iii
and in q9, I have problem applying the double angle identity in (a)
Can anyone pleease help?

Reply 1

Zacken

Original post by NachoPhilosopher

Hello everyone, I desperately need help in a paper I solved recently. Here are the two questions I need help in...

In q8, I have problem in finding least and greatest possible values in ii and iii
and in q9, I have problem applying the double angle identity in (a)
Can anyone pleease help?

What problem are you having? Show your working so far.

Reply 2

RDKGames

Study Forum Helper

Original post by NachoPhilosopher

Hello everyone, I desperately need help in a paper I solved recently. Here are the two questions I need help in...
In q8, I have problem in finding least and greatest possible values in ii and iii
and in q9, I have problem applying the double angle identity in (a)
Can anyone pleease help?

Been a while since I've done these.

Q8 least - You know

\arg(a+ai)=\frac{\pi}{4}

therefore

z-u=a+ai

and get

z

in terms of

a

. Then get it's magnitude and find the minimum of by using differentiation of z with respect to a. I'm sure you can do this quicker with an Argand diagram too by sketching it and finding the shortest distance from the origin to the loci.

Q8 greatest - Draw the Argand diagram of it. You can see that the point on the circle furthest away from the origin is one which is on the line that goes from the origin through the centre of the circle. You can work it out since you know the centre, so you can add on the horizontal and vertical distances onto it. You work those out from the right-angled triangle with hypotenuse 1 and angle

\frac{\pi}{4}

Q9 - Just set

8\cos^4(\theta)-3=1

and solve it.

(edited 7 years ago)

Reply 3

MathMoFarah

Original post by NachoPhilosopher

In q8, I have problem in finding least and greatest possible values in ii and iii

Sketch them first.

Original post by NachoPhilosopher

and in q9, I have problem applying the double angle identity in (a)
Can anyone pleease help?

Use De Moivre's identity thing:
cos(nx)=The real part of (cosx+isinx)^n

for i
then just make substitutions for a and b

Reply 4

MathMoFarah

Original post by RDKGames

Been a while since I've done these.

Q8 least - You know

\arg(a+ai)=\frac{\pi}{4}

therefore

z-u=a+ai

and get

z

in terms of

a

. Then get it's magnitude and find the minimum of by using differentiation of z with respect to a. I'm sure you can do this quicker with an Argand diagram too, but I'm not sure about it, just my thoughts.

Bear in mind when doing this that a is strictly greater than 0

Reply 5

RDKGames

Study Forum Helper

Original post by MathMoFarah

Bear in mind when doing this that a is strictly greater than 0

Yeah it has to be for the argument to hold true. Is there a pitfall with this method that you're referring to? :s-smilie:

Reply 6

MathMoFarah

Original post by RDKGames

Yeah it has to be for the argument to hold true. Is there a pitfall with this method that you're referring to? :s-smilie:

If you let a go negative from the Argand Diagram you can see a different solution is reached (which shouldn't be there) when you have a 'starting point' for the half line is in the top right diagonal 'half' of the Argand plane (aka when y>-x for a starting point x+yi)

(edited 7 years ago)

Reply 7

atsruser

Original post by NachoPhilosopher

Hello everyone, I desperately need help in a paper I solved recently. Here are the two questions I need help in...

In q8, I have problem in finding least and greatest possible values in ii and iii
and in q9, I have problem applying the double angle identity in (a)
Can anyone pleease help?

q8 ii): once you have sketched the locus of points satisfying the equation - it's a so-called open half-ray - then the answer should be visually obvious (though note that since the point

u

is not on locus, the word "least" in the question is slightly misleading, though that's being picky)

q8 iii): again sketch the locus defined by the equation, and work geometrically.

Reply 8

old_engineer

For 8(ii) the way to visualise the situation in terms of an Argand diagram is first to note that (z-u) must be a half-line starting at the origin and extending into the first quadrant at an angle pi/4 to the positive horizontal axis. We know that u=-3i, so (z-u)=(z+3i). (z+3i) is z transposed upwards by 3, so z must be a the half line parallel with (z-u) but starting at -3i rather than at the origin. The minimum value of mod(z) is the closest this line gets to the origin, which can be solved either by Pythagoras or using algebra and differentiating.