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 parties**constituting 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

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

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

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