You will need
- basic knowledge of trigonometry and geometry
Instruction
1
Calculate the angle, frictionangleand, if known to the other two angle α and β, as the difference between 180°−(α+β) as the sum of the angles ina square, e is always equal to 180°. For example, suppose there are two angle frictionangleand 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 frictionanglea 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 trianglea squareand 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 tregone 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 trianglea squareand, if you know the lengths of all three sides a, b and c tregon. To do this, calculate the area of triangleangleand 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 triangleangleand 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 trea 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°.
