Trigonometrical functions are elementary functions which arose when studying rectangular triangles. They express dependence of the parties of these figures on acute angles and a hypotenuse. The sine is direct trigonometrical function.
Instruction
1. If the considered triangle is rectangular, then use basic definition of trigonometrical function of a sine for acute angles which is considered as the relation of a leg opposite to this acute angle, to a hypotenuse of a rectangular triangle. You remember the following - the corner lying against a hypotenuse is always equal 90 °. And the sine of the angle in 90 ° is always equal to unit.
2. If the considered triangle is any, then to find value of a sine of the angle and, count value of a cosine of this corner. For this purpose use the theorem of cosines according to which a square are long one party has to be it is equal to a square of length of the second party plus a square of length of the third party minus the doubled work of the second and third parties increased by a cosine of the angle between the second and third party. For a triangle of KMN KM2=NM2+ NK2-2NM*NK*cosλ. From here count cosλ=KM2-NM2-NK22NM*NK I on formula sin2 λ=1-cos2 λ calculate sinλ=1-cos2λ
3. One more way of finding of a sine of the angle consists in use of two different formulas of the area of a triangle. One formula - in which only lengths of the parties of a triangle (Heron's formula) are involved. At you lengths of all parties of a triangle have to be known. Let's assume, the parties are equal to m, n, k Then use the following formula of Heron: S=p *p -n*p -k * (p △)-m), where poluperimetr triangle: n+k+m2=pA the second formula is the work of lengths of two parties and value of a sine of the angle between these parties: S (△) = n* k * sinµ.Т. to. the value S is identical, equate the right parts of formulas: * (p -m) = n*k * sinµ.И from this formula find p *p -n*p -k a sine of the angle of a which is opposite to the party With: sin µ=p *p -n*p -k * ksinusy other corners can be found (p -m) N* on the formulas similar to the last.