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How to find the sides of the triangle

How to find the sides of the triangle

To find the sides of the triangle, it is necessary to know the length of two sides and one angle value.

Or vice versa - the length of one side and the magnitude of the two angles.

The value of the third angle is easily calculated from the sum of the angles of a triangle equal 180 degrees.

instructions

1

According to the two sides and the angle between them
If you know the lengths of two sides of a triangle andthe angle between them, to find the length of a third party may be using the cosine theorem: the square of the length of the sides of a triangle equals the sum of the squares of the lengths of its two other sides minus twice the product of the sides of the cosine of the angle between them.

Hence we have:
c =? (a? + b? -2ab * cosC), where
a and b? the length of the known parties,

FROM ? the angle enclosed between the parties (opposing side required)

from ? the length of the unknown side.
Example 1.
Given a triangle with sides of 10 cm and 20 cm and an angle of 60 degrees between them. Find the length of the side.
Decision.
In the above formula, we get:
c =? (10? 20? -2 * 10 * 20 * cos60?) =? (500-200) =? 300 ~ 17.32
A: The length of the sides of the triangle opposite the side length 10 and 20 cm, and the size of the angle between them is 60? - ~ 17.32 cm.

2

In two corners and side
If you know the values ​​of the two angles and the length of onesides of the triangle, the length of the other two parties to find the most convenient using the theorem of sines: the ratio of the sines of angles of a triangle to the lengths of the opposite sides are equal.
sinA / a = sinB / b = sinC / c, where:
a, b, c? the length of the sides of a triangle, and A, B, C? value opposite angles.
Which the corners of the triangle are known? it does not matter, since, using the fact that the sum of the angles of a triangle is 180 degrees, you can easily find out the value of an unknown angle.
That is, for example, if we know the values ​​of the angles A and C and the side length a, the side length to be:
c = a * sinC / sinA

3

If the same source data necessary to find the length of the side b, then to use the theorem of sines, you need to know the angle B:
? Since B = -A-C 180, the side length b can be found by the formula:
b = a * sin (180? -A-C) / sinA
Example 2.
Let triangle ABC are known side length a = 10 cm and the angles A = 30 and C = 20. Find the length of the side b.
Solution: according to the formula derived above we get:
b = 10 * sin (180? -30? -20?) / sin30? = 10 * sin130? / 0,5 = 5 * sin130? ~ 3,83
A: The length of the sides of the triangle ~ 3.83 cm.

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