Advice 1: How to find the length of sides of isosceles triangle

Isosceles is a triangle in which the lengths of two sides are the same. To calculate the size of any of the parties need to know the length of the other sides and one angle or radius of the triangle circumscribed around the circle. According to the known values for calculations it is necessary to use the formulas derived from theorems of sine or cosine, or from the theorem on projections.
Instruction
1
If you know the length of base of isosceles triangle (A) and the value adjacent to it angle (angle between the base and any lateral face) (α), then calculate the length of each of the sides (B) based on the spherical law of cosines. It will be equal to the private from dividing the length of the base to twice the value of the cosine of the known angle B=A/(2*cos(α)).
2
The length of the sides of an isosceles triangle, which is its base (A) can be calculated on the basis of the same cosines, if you know the length of its lateral side (B) and the angle between it and the base (α). It will be equal to twice the product of the known sides into the cosine of the known angle A=2*B*cos(α).
3
Another way to find the length of the base of an isosceles triangle can be used if we know the value of him opposite the angle (β) and the length of the lateral side (B) of the triangle. It will be equal to twice the product of length of the lateral sides by the sine of half the value of the known angle A=2*B*sin(β /2).
4
Similarly, we can deduce the calculation formula of lateral sides of the isosceles triangle. If the known base length (A) and the angle between the equal sides (β), the length of each (B) will be equal to the result of dividing the base length to twice the sine of half the value of the known angle B=A/(2*sin(β /2)).
5
If you know the radius described around an isosceles triangle on the circle (R), then the lengths of its sides can be calculated, knowing the value of one of the corners. If we know the value of the angle between the sides (β), the length of the partiesconstituting the grounds (A), is equal to twice the product of the radius of the circumscribed circle by the sine of this angle A=2*R*sin(β).
6
If you know the radius of the circumscribed circle (R) and the angle adjacent to the base (α), the length of the lateral side (B) will be equal to twice the product of the base length by the sine of the known angle B=2*R*sin(α).

Advice 2: How to find sides of triangle

To find the sides of the triangle, you must know the lengths of two sides and the size of one angle. Or Vice versa - the length of one side and the values of the two angles. The value of the third angle is easily calculated from the equality of the sum of the angles of a triangle is 180 degrees.
Instruction
1
Two sides and angle between them
If you know the lengths of two sides of a triangle and the angle between them, then to find the length of the third side using the theorem of cosines: 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 these sides into the cosine of the angle between them.

Hence we have:
C=√(A2+b2-2аb*cosC), where
a and b are the lengths of the known sides

With the size of the angle between these sides (opposite the side)

– the length of the required side.
Example 1.
Given a triangle with sides of 10 cm and 20 cm and the angle between them is 60 degrees. Find the length of a side.
Solution.
According to the above formula we get:
with=√(102+202-2*10*20*cos60º)=√(500-200)=√300~17,32
Answer: the length of a side of the triangle opposite the sides with lengths 10 and 20 cm and the size of the angle between them is 60º - ~17,32 see
2
According to two angles and the side
If the known values of two angles and the length of one of the sides of a triangle, the lengths of the other two sides the easiest to find using the theorem of sines: the ratio of the sines of the angles of a triangle to the lengths of opposite sides are equal.
sinA/a=sinB/b=sinC/C where:
a, b, c the lengths of the sides of a triangle, and A, B, C – values of the opposite corners.
What are the angles of a triangle is known – is not important, since, using the fact that the sum of the angles of a triangle equal 180 degrees, you can easily find out the value of the unknown angle.
That is, for example, if the known values of angles A and C and side length a, side length s will be:
C=a*sinC/sinA
3
If the same source data, you need to find the length of side b, then to use the theorem of sines, we need to know the size of this angle:
since B=180º-A-C, the length of side b can be found by the formula:
b=a*sin(180º-A-C)/sinA
Example 2.
Let in triangle ABC the known side length a=10 cm and angles A=30 and C=20. To find the length of side b.
Solution: in the above formula we get:
b=10*sin(180º-30º-20º)/sin30º=10*sin130º/0,5=5*sin130º~3,83
Answer: the length of a side of a triangle is ~3,83 see
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