You will need

- basic knowledge of trigonometry and geometry

Instruction

1

Calculate the angle, friction

**angle**and, if known to the other two angle α and β, as the difference between 180°−(α+β) as the sum of the angles in**a square**, e is always equal to 180°. For example, suppose there are two angle friction**angle**and α=64°, β=45°, then the unknown angle γ=180−(64+45)=71°.2

Use the theorem of cosines when you know the lengths of two sides a and b friction

**angle**a and angle α between them. Find the third side by the formula c=√(a2+b2−2*a*b*cos(α)), as the squared length of any side of triangle**a square**and is equal to the sum of the squares of the lengths of the other sides minus twice the product of the lengths of these sides into the cosine of the angle between them. Write down the theorem of cosines for the other two sides: a2=b2+c2−2*b*c*cos(β), b2=a2+c2−2*a*c*cos(γ). Express of these formulas, the unknown angles: β=arccos((b2+c2−a2)/(2*b*c)), γ=arccos((a2+c2−b2)/(2*a*c)). For example, suppose tre**gon**e well-known side a=59, b=27, the angle between them α=47°. Then the unknown side c=√(592+272-2*59*27*cos(47°))≈45. Then β=arccos((272+452-592)/(2*27*45))≈107°, γ=arccos((592+452-272)/(2*59*45))≈26°.3

Find the angles of triangle

**a square**and, if you know the lengths of all three sides a, b and c tre**gon**. To do this, calculate the area of triangle**angle**and by Heron's formula: S=√(p*(p−a)*(p−b)*(p−c)), where p=(a+b+c)/2 – properiter. On the other hand, since the area of triangle**angle**and is equal to S=0,5*a*b*sin(α), we Express from this formula the angle α=arcsin(2*S/(a*b)). Similarly, β=arcsin(2*S/(b*c)), γ=arcsin(2*S/(a*c)). For example, suppose that we are given tre**a square**with sides a=25, b=23 and C=32. Then count properiter p=(25+23+32)/2=40. Calculate the area by Heron's formula: S=√(40*(40-25)*(40-23)*(40-32))=√(40*15*17*8)=√(81600)≈286. Find the angles: α=arcsin(2*286/(25*23))≈84°, β=arcsin(2*286/(23*32))≈51°, and the angle γ=180−(84+51)=45°.