Advice 1: How to find the side of a triangle knowing the side and the angle

In the General case, knowledge of the length of one side and one angle of a triangle is insufficient to determine the length of the other side. These data may be sufficient to determine the sides of a right triangle, and isosceles triangle. In the General case what you need is another parameter of the triangle.
How to find the side of a triangle knowing the side and the angle
You will need
  • The sides of the triangle, the angles of a triangle
For starters, you can consider special cases and start with the case of a right triangle. If you know that the triangle is rectangular and known for one of its acute angles, then the length of one of the parties can be found and other sides of the triangle.

To find the length of the other sides you need to know which side of the specified triangle is the hypotenuse or any of the other two sides. The hypotenuse lies against the straight edge, the legs form a right angle.

Consider right triangle ABC with right angle ABC. Imagine you are given its hypotenuse and AC, for example, the acute angle BAC. Then the other two sides of the triangle are equal: AB = AC*cos(BAC) (adjacent side to angle BAC), BC = AC*sin(BAC) (the side opposite the angle BAC).
Let now set to the same angle BAC, for example, the side AB. Then AC is the hypotenuse of this right triangle is equal to: AC = AB/cos(BAC) (respectively, AC = BC/sin(BAC)). The other leg BC is given by BC = AB*tg(BAC).
Another special case - if the triangle ABC is isosceles (AB = AC). Imagine you are given the base BC. If you specify the angle BAC, then the sides AB and AC can be calculated using the following formula: AB = AC = (BC/2)/sin(BAC/2).

If you specify the base angle ABC or ACB, then AB = AC = (BC/2)/cos(ABC).
Let set one of the sides AB or AC. If you know the angle BAC, BC = 2*AB*sin(BAC/2). If you know the angle ABC or the angle ACB at the base, BC = 2*AB*cos(ABC).
You can now consider the General case of a triangle when length of one side and one angle is not enough to find the length of the other side.

Let in triangle ABC the side AB is set to one of the adjoining corners, for example, angle ABC. Then, knowing the side BC, by theorem of cosines to find the side AC. It will be equal to: AC = sqrt (AB^2)+(BC^2)-2*AB*BC*cos(ABC))
Now let the given side AB and the opposite angle ACB to her. May also known, for example, angle ABC. By theorem of sines, AB/sin(ACB) = AC/sin(ABC). Hence, AC = AB*sin(ABC)/sin(ACB).

Advice 2: How to find the side of a triangle

Side of the triangle is a direct, limited its vertices. All of them have figures of three, this number determines the number of almost all graphics characteristics: angle, midpoint, bisectors, etc. to find the side of the triangle, you should carefully examine the initial conditions of the problem and determine which of them may be basic or intermediate values to calculate.
How to find the side of a triangle
The sides of the triangle, like other polygons have their own names: the sides, base, and hypotenuse and legs of the figure with a right angle. This facilitates the calculations and formulas, making them more obvious even if the triangle is arbitrary. The figure of graphics, so it is always possible to arrange so as to make the solution of the problem more visible.
Sides of any triangle are connected and the other characteristics of the various ratios that help calculate the required value in one or more actions. Thus the more complex the task, the longer the sequence of steps.
The solution is simplified if the triangle is standard: the words "rectangular", "isosceles", "equilateral" immediately allocate a certain relationship between its sides and angles.
The lengths of the sides in a right triangle are connected by Pythagorean theorem: the sum of the squares of the legs equals the square of the hypotenuse. And the angles, in turn, are associated with the parties to the theorem of sines. It affirms the equality relations between the lengths of the sides and the trigonometric function sine of the opposite angle. However, this is true for any triangle.
Two sides of an isosceles triangle are equal. If their length is known, it is enough only one value to find the third. For example, suppose we know the height held to it. This cut divides the third side into two equal parts, and allocates two rectangular triangleH. Considered one of them, by the Pythagorean theorem find the leg and multiply it by 2. This will be the length of an unknown side.
Side of the triangle can be found through other sides, corners, length, altitude, median, bisector, perimeter size, area, inradius, etc. If you can't apply the same formula to produce a series of intermediate calculations.
Consider an example: find side of an arbitrary triangle, knowing the median ma=5, held for her, and the lengths of the other two medians mb=7 and mc=8.
Resented involves the use of formulas for the median. You need to find the way . Obviously, there should be three equations with three unknowns.
Write down the formulae for the medians:ma = 1/2•√(2•(b2 + c2) – a2) = 5;mb = 1/2•√(2•(a2 + c2) – b2) = 7;mc = 1/2•√(2•(a2 + b2) – c2) = 8.
Express c2 from the third equation and substitute it into the second:c2 = 256 – 2•a2 – 2•b2 b2 = 20 → c2 = 216 – a2.
Lift both sides of the first equation in the square and find a by entering explicit values:25 = 1/4•(2•20 + 2•(216 – a2) – a2) → a ≈ 11,1.
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