Triangle call the part of the plane limited to three pieces of the straight lines (parties of a triangle) having in pairs on one general end (triangle tops). Corners of a triangle can be found according to the Theorem of the sum of corners of a triangle.
Instruction
1. The theorem of the sum of corners of a triangle says that the sum of corners of a triangle is 180 °. Let's consider several examples tasks with the different set parameters. First, let two corners α = 30 °, β = 63 ° are set. It is necessary to find the third corner γ. We find it directly from the theorem of the sum of corners of a triangle: α + β + γ = 180 ° => γ = 180 ° - α - β = 180 ° - 30 ° - 63 ° = 87 °.
2. Now we will consider a problem of finding of the third corner of a triangle of more general view. Let to us three parties of a triangle |AB| = be known to a, for |BC| = b, |AC| = c. Also needs to find three corners α, β and γ. Let's use the theorem of cosines for finding of a corner β. According to the theorem of cosines the square of the party of a triangle is equal to the sum of squares of two other parties minus the doubled work of these parties and the cosine of the angle concluded between them. I.e. in our designations c^2 = a^2 + b^2 – 2 * a * b * cos β => cos β = (a^2 + b^2 - c^2) / (2 * a * b).
3. Further we will use the theorem of sine for finding of a corner α. According to this theorem of the party of a triangle are proportional to sine of opposite corners. Let's express from this ratio a sine of the angle α: a/sin α = b/sin β => sin α = b * sin β/a. The third corner is found according to the theorem already known to us of the sum of corners of a triangle on a formula γ = 180 ° - (α + β).
4. Let's give an example of the solution of a similar task. Let the parties of a triangle of a = 4 be given, to b = 4 * √2, by c = 4. From a condition we see that it is an isosceles rectangular triangle. I.e. as a result we have to receive corners 90 °, 45 ° and 45 °. Let's count these corners on the way given above. According to the theorem of cosines we find a corner β: cos β = (16 + 32 - 16) / (2 * 16 * √2) = 1/√2 = √2/2 => β = 45 °. Further we find a corner α according to the theorem of sine: sin α = 4 * √2 * √2 / (2 * 4) = 1 => α = 90 °. And at last, having applied the theorem of the sum of corners of a triangle, we receive a corner γ = 180 ° - 45 ° - 90 ° = 45 °.