To find the sides of a triangle, you need to know the lengths of the two sides and the size of one corner.
Or vice versa - the length of one side and the size of two angles.
The magnitude of the third angle can be easily calculated from the equality of the sum of the angles of the triangle to 180 degrees.
On two sides and the corner between them
If the lengths of the two sides of the triangle are known andThe size of the angle between them, then you can find the length of the third side by using the cosine theorem: the square of the length of the side of the triangle is equal to the sum of the squares of the lengths of its two other sides minus the doubled product of these sides by the cosine of the angle between them.
Hence we have:
C =? (A? + B? -2ab * cosC), where
A and b? Length of known parties,
FROM ? The value of the angle between these sides (opposite the target side),
from ? Length of the desired side.
A triangle with sides of 10 cm and 20 cm and an angle between them equal to 60 degrees is given. Find the length of the side.
By the above formula, we get:
C =? (10? + 20? -2 * 10 * 20 * cos60?) =? (500-200) =? 300 ~ 17.32
Answer: the length of the side of the triangle opposite the sides of lengths of 10 and 20 centimeters and the value of the angle between them is 60? ~ 17.32 cm.
On two corners and side
If the values of two angles and the length of oneSides of the triangle, then the length of the other two sides is most conveniently found using the sine theorem: the ratio of the sines of the angles of the triangle to the lengths of the opposite sides is equal to each other.
SinA / a = sinB / b = sinC / c, where:
A, b, c? The length of the sides of the triangle, and A, B, C? The values of the opposite angles.
Which angles of the triangle are known? It is not important, since, taking advantage of the fact that the sum of the angles of the triangle is 180 degrees, you can easily find out the value of the unknown angle.
That is, for example, if the angles A and C are known and the length of the side a is known, then the length of side c will be:
C = a * sinC / sinA
If for the same initial data it is necessary to find the length of the side b, then to use the sine theorem, you need to know the value of the angle B:
Since B = 180? -A-C, then the length of the side b can be found from the formula:
B = a * sin (180? -A-C) / sinA
Let the side length a = 10 cm and the angles A = 30 and C = 20 be known in the triangle ABC. Find the side length b.
Solution: according to the above formula, we get:
B = 10 * sin (180? -30? -20?) / Sin30? = 10 * sin130? / 0.5 = 5 * sin130? ~ 3.83
Answer: the length of the side of the triangle is ~ 3.83 cm.