Cosine of the angle is called the relation of a leg, adjacent to this corner, to a hypotenuse. This size, as well as other trigonometrical ratios, is used for the decision not only rectangular triangles, but also many other tasks.
Instruction
1. For any triangle with tops And, In and With a problem of finding of a cosine it is identical to all three corners if an acute triangle. If in a triangle there is an obtuse angle, definition of its cosine should be considered separately.
2. In an acute triangle with tops And, In and With find a cosine of the angle at A. Opustite's top height from top In on the party of a triangle the EXPERT. Height point of intersection with the party the EXPERT designate D and consider a rectangular triangle of ABD. In this triangle the party of AV of an initial triangle is a hypotenuse, and legs — height of BD of an initial acute triangle and the piece of AD belonging to the party the EXPERT. The cosine of the angle And is equal to AD/AB relation as the leg of AD is adjacent to a corner And in a rectangular triangle of ABD. If it is known in what ratio height of BD divides the party the EXPERT of a triangle, then the cosine of the angle And is found.
3. If the size AD is not given, but BD height is known, the cosine of the angle can be defined through its sine. The sine of the angle And is equal to the relation of height of BD of an initial triangle to the party the EXPERT. The main trigonometrical identity establishes connection between a sine and a cosine of the angle: Sin² A+ Cos² A=1. For finding of a cosine of the angle And calculate: 1-(BD/AC)², it is necessary to take a square root from the received result. The cosine of the angle And is found.
4. If in a triangle all parties are known, then find a cosine of any corner according to the theorem of cosines: the square of the party of a triangle is equal to the sum of squares of two other parties without the doubled work of these parties on a cosine of the angle between them. Then a cosine of the angle And in a triangle with the parties and, b, with calculate on a formula: Cos A = (and²-b²-c²) / 2*b*с.
5. If in a triangle it is necessary to define a cosine of an obtuse angle, use a reduction formula. The obtuse angle of a triangle is more direct, but it is less developed, it can be written down as 180 °-α where α — the acute angle supplementing an obtuse angle of a triangle to developed. On a formula of reduction find a cosine: Cos (180 °-α) = Cos α.