How to find corners when lengths of the parties of a triangle are known

How to find corners when lengths of the parties of a triangle are known

Sizes of the corners lying in triangle tops and lengths of the parties forming these tops are connected among themselves by certain ratios. These relations are expressed most often through trigonometrical functions - generally through a sine and a cosine. Knowledge of lengths of all parties of a figure is enough that with use of these functions to restore sizes of all three corners.


1. For calculation of size of any of corners of any triangle use the theorem of cosines. It says that a square of length of any party (for example, A) it is equal to the sum of squares of lengths of two other parties (B and C) from which the work of their lengths on a cosine of the angle is subtracted (α), lying in the top formed by them. It means that you can express a cosine through lengths of the parties: cos(α) = (B²+C²-A²) / (2*A*B). To receive the size of this corner in degrees, apply the return to the received expression to a cosine function - an arccosine: α = arccos ((B²+C²-A²) / (2*A*B)). In such a way you calculate the size of one of corners - in this case that which lies opposite to the party And.

2. For calculation of two remained corners it is possible to use the same formula, interchanging the position in it of lengths of the known parties. But simpler expression with smaller number of mathematical operations can be received, having involved other postulate from the field of trigonometry - the theorem of sine. She claims that the relation of length of any party to a sine opposite a corner are equal to it in a triangle. It means that you can express, for example, a sine of the angle β, B lying opposite to the party through length of the party of C and already calculated corner α. Increase B length by a sine α, and divide result into length of C: sin(β) = B*sin(α)/C. As well as in the previous step, calculate the size of this corner in degrees with use of the inverse trigonometrical function - this time an arcsine: β = arcsin (B*sin(α) / C).

3. The size of the remained corner (γ) can be calculated on any of the formulas received in the previous steps, having traded in them places of length of the parties. But it is simpler to involve one more theorem - about the sum of corners in a triangle. She claims that this sum is always equal 180 °. As two of three corners are already known to you, just take away from 180 ° their sizes to receive the size of the third: γ = 180 °-α-β.

Author: «MirrorInfo» Dream Team