crazy (1 Viewer)

elliottjohnson · Jun 24, 2009

question for u math gods.

I thought this question was a little crazy.
does anyone have any ideas?

By using the t-formulae, show that the equation a sin x + b cos x = c

has real roots if a^2 + b^2 >= c^2

Aquawhite · Jun 24, 2009

Using the t-formulae? No thankyou...

Timothy.Siu · Jun 24, 2009

elliottjohnson said:
question for u math gods.

I thought this question was a little crazy.
does anyone have any ideas?

By using the t-formulae, show that the equation a sin x + b cos x = c

has real roots if a^2 + b^2 >= c^2

uhhh t formula?
i'm sure its just the same as transformations...

u get
root(a^2+b^2)sin(x+pi/4)=c
sin(x+pi/4)=c/root(a^2+b^2)
-1<=sin @<=1
therefore for real roots a^2 + b^2 >= c^2

annabackwards · Jun 24, 2009

I've uploaded my solution which i did using the only pen i could find - a brown/gold texta XD

Click here for solution

It does use the t formula and i hope it makes sense

Aquawhite · Jun 24, 2009

annabackwards said:
I've uploaded my solution which i did using the only pen i could find - a brown/gold texta XD

Click here for solution

It does use the t formula and i hope it makes sense

Crazy, and excellent job! I could not have been bothered to use such brain power to work that out... and probably wouldn't have been able too (especially since trigonometry is my weakest topic)

Drongoski · Jun 24, 2009

I'll give it a go.

$\80dpi Let\ tan\frac{x}{2}\ =\ t\ \ \Rightarrow \ \ sin\ x =\ \frac{2t}{1+t^{2}}\ \ \ \ cos\ x=\ \frac{1-t^{2}}{1+t^{2}}\\ \\ \therefore \ \ asinx\ +\ bcosx\ =\ c\ \ \Rightarrow \ \ a[\frac{2t}{1+t^{2}}]\ +\ b[\frac{1-t^{2}}{1+t^{2}}]\ =\ c\\ \\ so\ that:\ \ 2at\ +\ b(1-t^{2})\ =\ c(1+t^{2})\ \ \Rightarrow \ \ (c+b)t^{2}\ -\ 2at\ +\ (c-b)\ =\ 0\\ \\ Quad\ eqn\ \ has\ real\ roots \ \Leftrightarrow \ \ \Delta \ = \ \ \ <b^{2}-4ac> \\ \\ \ \ \ =\ (-2a)^{2}-4(c+b)(c-b)\ \geq \ 0\\ \\ \ i.e. \ \ \Leftrightarrow \ \ 4a^{2}+4b^{2}\ \geq \ 4c^{2}\ \ \ \ i.e.\ \ \ a^{2}\ +\ b^{2}\ \geq \ c^{2}\\ \\ QED$

annabackwards · Jun 24, 2009

Aquawhite said:
Crazy, and excellent job! I could not have been bothered to use such brain power to work that out... and probably wouldn't have been able too (especially since trigonometry is my weakest topic)

I'm glad that the solution made sense, i was worried that i'd be quite rusty after not doing trig for more than 1/2 a year ^^ I was only bothered because i'm "constructively procrastinating" >.>

But if trig is your weakest topic, just do lots of examples and it'll slowly sink in

Aquawhite · Jun 24, 2009

annabackwards said:
I'm glad that the solution made sense, i was worried that i'd be quite rusty after not doing trig for more than 1/2 a year ^^ I was only bothered because i'm "constructively procrastinating" >.>

But if trig is your weakest topic, just do lots of examples and it'll slowly sink in

It was easy enough to understand

I need to practice lots of everything before I face the trials and eventually the end HSC exams. I want to really have everything perfected before the trials in Week 3 next term

... not long. But the holidays will give me ample time.

annabackwards · Jun 24, 2009

Aquawhite said:
It was easy enough to understand

I need to practice lots of everything before I face the trials and eventually the end HSC exams. I want to really have everything perfected before the trials in Week 3 next term ... not long. But the holidays will give me ample time.

Good luck, you'll be fine

Aquawhite · Jun 24, 2009

Liking the comfort in here!

elliottjohnson · Jun 25, 2009

thanks very much guys!!

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