The triangle is defined by its corners and the parties. As corners allocate acute triangles – all three corners sharp, obtusangular – one obtuse angle, rectangular – one right angle, all corners are equal in an equilateral triangle to 60. It is possible to find a triangle corner in different ways depending on basic data.
It is required to you
- basic knowledge of trigonometry and geometry
Instruction
1. Calculate a triangle corner if two other corners α and β as a difference 180 ° − (α+β are known) as sum corners is always equal in a triangle 180 °. For example, let two corners of a triangle α=64 °, β=45 °, then an unknown corner γ=180− (64+45)=71 ° are known.
2. Use the theorem of cosines when lengths of two parties of an and b of a triangle and a corner α between them are known. Find the third party on a formula c= √ (a²+b²−2*a*b*cos(α)) as the square of length of any party of a triangle is equal to the sum of squares of lengths of other parties minus the doubled work of lengths of these parties on a cosine of the angle between them. Write down the theorem of cosines for two other parties: a²=b²+c²−2*b*c*cos(β), b²=a²+c²−2*a*c*cos(γ). Express unknown corners from these formulas: β=arccos ((b²+c²−a²) / (2*b*c)), γ=arccos ((a²+c²−b²) / (2*a*c)). For example, let in a triangle the parties of a=59, b=27, a corner between them α=47 ° are known. Then unknown party of c= √ (59²+27²−2*59*27*cos (47 °)) ≈45. Means β=arccos ((27²+45²−59²)/(2*27*45))≈107 °, γ=arccos ((59²+45²−27²)/(2*59*45))≈26 °.
3. Find triangle corners if lengths of all three parties of a, b and c of a triangle are known. For this purpose calculate the area of a triangle on Heron's formula: S= √ (p * (p−a) * (p−b) * (p−c)), where p=(a+b+c)/2 – poluperimetr. On the other hand, as the area of a triangle is equal to S=0.5*a*b*sin(α), express from this formula a corner α=arcsin (2*S / (a*b)). Similarly, β=arcsin (2*S / (b*c)), γ=arcsin (2*S / (a*c)). For example, let the triangle with the parties of a=25, b=23 and with =32 is given. Then count poluperimetr p=(25+23+32)/2=40. Calculate the area on Heron's formula: S= √ (40*(40−25)*(40−23)*(40−32))= √ (40*15*17*8)= √ (81600) ≈286. Find corners: α=arcsin (2*286/(25*23))≈84 °, β=arcsin (2*286/(23*32))≈51 °, and corner γ=180− (84+51)=45 °.