Trig problems... (1 Viewer)

[sladehk]

Heya! I was having problem with these questions:

1) Find the exact value of sin61
2) Show that cos46 = (root2/360)*(180-Pi)

Thanks

 

[who_loves_maths]

err... are you sure you where just given those two question like that?
no precursors or additional bits that lead you into the question?
(seems a bit incredulous to me)

 

[maths > english]

based on the answer to question 2 you're not looking for an exact value of cos 46, just a way to get a very close approximation

let y = cos x

^dy/_dx = -sin x

because ^Δy/_Δx = -sin x

Δy = -sin x * Δx

so y+Δy can be approximated by cos x - sin x * Δx

46 degrees = ^46π/₁₈₀ rads

cos ^46π/₁₈₀ = cos ^45π/₁₈₀ - sin ^45π/₁₈₀ * ^π/₁₈₀ (approximately)

¹/_√2 - ¹/_√2 * ^π/₁₈₀

¹/_√2 * (1 - ^π/₁₈₀)

^√2/₂ * (^{180 - π}/₁₈₀)

^√2/₃₆₀ * (180 - π)

= 0.694765439691663173714689137134 (approx)

cos 46 = 0.694658370458997286656406299422 (approx)

 

[maths > english]

answering the first question you can get a close approximation to sin 61 in a similar way

61 degrees = 61π/180 rads

sin 61π/180 = sin 60π/180 + cos 60π/180 * π/180 (approximately)

√3/2 + 1/2 * π/180

1/360 * (180√3 + π)

= 0.874752050044410294648341624597 (approx)

sin 61 = 0.874619707139395800284636958662 (approx)

 

[sladehk]

err... are you sure you where just given those two question like that?
no precursors or additional bits that lead you into the question?
(seems a bit incredulous to me)

yea we were... and it requires an "exact" for the 1st and a "show"/proof of the second... thnx anywayz!

 

[who_loves_maths]

^ what sort of technique is that maths>english?

 

[maths > english]

its an approximation technique taught in math1141 at unsw:

http://www2.maths.unsw.edu.au/ForStudents/courses/math3241/calculus/hcalchap3_PorTcd2.pdf

look at page 12

 

[who_loves_maths]

^oh ic, thx.
is math1141 like an advanced maths class? or just a normal one?

Edit: why not just use (since we are only talking about high school maths here) approximation methods taught in 3u: Newton's Method

Sin(61) is a zero of F(x) = ArcSin(x) - (61pi)/180 ;
so x(2) = x(1) - (ArcSinx - (61pi)/180)(Sqrt(1 - x^2)) ...

i know it's not as simple as your method, and it takes longer, and it also requires the use of a calculator... but the thing is, if the initiall question is only after an approximation, then why not just use your calculator in the first place? why bother with either methods?

 

[maths > english]

the course name for math1141 is higher maths 1A, its basically math1131 (maths 1A) in greater depth. see:

http://www.handbook.unsw.edu.au/undergraduate/courses/2005/MATH1141.html

 

[maths > english]

i guess the only point would be to express sin 61 in a form that gives a clearer idea of its value.

e.g. on observation u could say √3 = about 1.7, pi about 3.1 which gives an idea of the value of 1/360 * (180√3 + π)

 

[who_loves_maths]

Originally Posted by sladehk
1) Find the exact value of sin61
2) Show that cos46 = (root2/360)*(180-Pi)

in question 2), if cos46 = (root2/360)*(180-Pi) is indeed the exact value of cos46, then cos46 is a transcendental number (because of the 'pi'). which means you cannot deduce that exact value by forming and solving polynomials of any degree with rational coefficients.
the problem here is that usually in high school, and in particular in 4u maths, we are asked to find exact values of numbers like cos(5pi/12) by forming simple quadratic equations with rational coefficients and then solving them (eg. in the Complex Numbers topic, we do this often).
but since that case is no long a possibility due to the transcedence of cos46, then you probably can't find the exact value of cos46 using conventional 4u techniques. you can, however, obtain approximations using 3u techniques.

are you sure those two questions are only high school level maths???

also, for question 1:
sin61 = sin(46 +15) = sin(15)cos46 + cos(15)sin46 ;
but sin(15) and cos(15), i know for a fact, are not trascendental and have exact values in terms of root extractions, etc...
but since cos46 and sin46 are transcendental, then clearly sin61 is also a transcendental number... so it meet the same problem as that of cos46, you can't find its exact value by forming and solving polynomials.

 

[maths > english]

cos 46 = ^√2/₃₆₀ * (180 - π) was obtained using an approximation technique and is therefore not the exact value

^√2/₃₆₀ * (180 - π) = 0.694765439691663173714689137134 (approx)

cos 46 = 0.694658370458997286656406299422 (approx)

as you can see the values are not the same after the 3rd decimal place

 

[who_loves_maths]

Originally Posted by maths > english
cos 46 = √2/360 * (180 - π) was obtained using an approximation technique and is therefore not the exact value

√2/360 * (180 - π) = 0.694765439691663173714689137134 (approx)

cos 46 = 0.694658370458997286656406299422 (approx)

as you can see the values are not the same after the 3rd decimal place

hence i said:

Originally Posted by who_loves_maths
in question 2), if cos46 = (root2/360)*(180-Pi) is indeed the exact value of cos46, then

see the 'if'?

 

[maths > english]

yea we were... and it requires an "exact" for the 1st and a "show"/proof of the second... thnx anywayz!

sorry <i>who_loves_maths</i>, i was responding to what <i>sladehk</i> said before, not what you said

just wanted to make clear that these questions had nothing to do with finding exact values, showing or proving anything

 

[who_loves_maths]

Originally Posted by maths > english
sorry who_loves_maths, i was responding to what sladehk said before, not what you said

just wanted to make clear that these questions had nothing to do with finding exact values, showing or proving anything.

oh okay, sorry about that then, misunderstanding on my part