One useful trigonometric formula can be obtained by expressing the area of an arbitrary triangle with sinus.
If one then proceeds to divide through by abc/2 one gets the Sinustheorem.
sinA/a = sinB/b= sinC/c
which is a relation between the sinus of the angle A and the side a standing opposite to angle A. By permutating the angles and sides you get the other two relations.
Another important formula is the cosinetheorem:
c2 = a2+b2 – 2ab cos(c)
.
I angle a=90 this reduces to the Pythagoren theorem.
A beautiful proof of this formula can be found with the aid of vectoranalysis:
compute (a+b)(a+b) = a2 +b2 + 2 a*b.
The last term is a scalar product between two vectors
a×b×cos(180-c) = -a×b×cos(c).
This gives the cosinetheorem:
c2 = a2+b2 – 2ab cos(c) Q.E.D.
Find the value of x
Övning 1410 i Sjunnesson: en storks näbb är 26 cm lång. Hur stora matbitar kan den äta om den maximalt kan kan öppna munnen 41 grader?
Cosinussatsen ger:
x^2 = 26^2 + 26^2 – 2 26^2 *cos41 ×
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