T formula question help (1 Viewer)

[EpikHigh]

Hey guys, wondering if someone could show me the steps of sovling this, domain is theta is greater than or equal to zero, less than or equal to 360

 

[Drongoski]

Similarly you can work out for t = -5

 

[Sanjeet]

You can also use the auxiliary angle method unless the question specifies t formula

 

[Leffife]

sin θ = (2t) / (1 + t²)

cos θ = (1 - t²)/(1 + t²)

Just substitute into the equation.

2((2t) / (1 + t²)) - 3((1 - t²)/(1 + t²)) = 2
4t/(1 + t²) - 3 + t²/(1 + t²) = 2
(t² + 4t - 3)/(1+t²) = 2
t² + 4t - 3 = 2t² + 2
t² - 4t + 5 = 0
Using quadratic formula: t = (4 ± √12)/2

Then you know that tan(θ/2) = t

Afterwards, you can continue and find θ.

 

[EpikHigh]

Similarly you can work out for t = -5

Thanks man! I get it now Except I don't get the last step kind of, since I haven't learned radians :L

 

[Leffife]

sin θ = (2t) / (1 + t²)

cos θ = (1 - t²)/(1 + t²)

Just substitute into the equation.

2((2t) / (1 + t²)) - 3((1 - t²)/(1 + t²)) = 2
4t/(1 + t²) - 3 + t²/(1 + t²) = 2
(t² + 4t - 3)/(1+t²) = 2
t² + 4t - 3 = 2t² + 2
t² + 4t - 5 = 0
(t + 5)(t - 1) = 0
t = - 5 or t = 1

We know that tan(θ/2) = t

Thus, let tan(θ/2) = 1
Thus θ/2 = π/4
And just multiply it by '2', i.e. θ = 2π/4 = π/2

 

[Leffife]

Thanks man! I get it now Except I don't get the last step kind of, since I haven't learned radians :L

180 degrees = π (pi)
360 degrees = 2π (2 meat pies )

 

[EpikHigh]

You can also use the auxiliary angle method unless the question specifies t formula

Yeah I would have probably used auxiliary method but the question did specify using the t formula.

 

[Sanjeet]

Thanks man! I get it now Except I don't get the last step kind of, since I haven't learned radians :L

That's pretty much all there is to learn for radians

 

[Nooblet94]

Similarly you can work out for t = -5

tut tut, forgot to test for theta=180

 

[Drongoski]

tut tut, forgot to test for theta=180

Oh dear. Must retire from tutoring!

 

[Fus Ro Dah]

Oh dear. Must retire from tutoring!

In all honesty Drongoski, I think the whole idea of "Showing that you test 180 or else you lose marks" is very silly. It can very be easily seen and instead of deducting perfectly good students for it, the Examination Committee could simply include questions where 180 is indeed a solution, and those who don't check it get penalised for missing it out, as opposed to just penalising people for not actually writing it down when it can trivially be seen by simple inspection.