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[Math] Trigonometry puzzle
Message de TravisKidd posté le 26-02-2007 à 08:42:47 (S | E | F | I)
Here's a little trigonometry puzzle you might like:

I. Using a calculator, calculate sin 7°, sin 53°, and sin 67°. Compare these values. What relation does there appear to be among them?

II. Do the same thing for sin 23°, sin 37°, and sin 83°.

III. Based on your discoveries in I. and II., hypothesize a trigonometric identity.

IV. Prove this identity using the following well-known fact and identities:
cos 60° = 1/2
sin(-x) = -(sin x)
cos(-x) = cos x
sin(x+y) = (sin x)(cos y) + (cos x)(sin y)

V. Give a specific example of this identity using three specific values. (Two such examples are given in I. and II.) Show using a calculator that this identity applies to your three values.

Good luck! I will post the correction here (in this thread) in approximately one week.

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Modifié par bridg le 17-03-2007 15:56
transfert en maths

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Réponse: [Math] Trigonometry puzzle de magstmarc, postée le 27-02-2007 à 08:56:31 (S | E)
Hi Travis,

I. sin 7° is about 0,12
sin 53° is about 0,80
sin 67° is about 0,92

We can conjecture (does that word exist ?) that sin 67° = sin 53° + sin 7°

II. sin 23° is about 0,39
sin 37° is about 0,60
sin 83° is about 0,99

We can hypothetise that sin 83° = sin 37° + sin 23°

III. Conjecture : for every real number x, in degrees,
sin(60° + x)= sin(60° - x)+ sin x

IV. x being a real number, in degrees,
sin(60°+x)= sin60°cosx + cos60°sinx
and
sin(60°-x)= sin[60° + (-x)]
= sin60°cos(-x) + cos60°sin(-x)
= sin60°cosx - cos60°sinx

If we substract these two identities we get
sin(60°+x) - sin(60°-x) = 2cos60°sinx
= 2.(1/2).sin x
= sin x
that is,
sin(60°+x) = sin(60°-x) + sin x

V. So if x = 5°, sin 65° = sin 55° + sin 5°
Calculator :
sin 5° is about 0,09
sin 55° is about 0,82
sin 65° is about 0,91

Nice exercise Traviskidd

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