Good day, dear guests and subscribers!
Today I would like to highlight the theme of the use of trigonometry to the construction, since mathematics is very closely to this area.
The angles and sides of any whether it is isosceles, equilateral triangle or versatile communicate between a certain trigonometric relationships, the main of which is isolated as the "law of sines" and "theorem cosine ".
Due to the great mathematician of ancient times, a formula allowing for three elements of any triangle - to restore the other three!
Next, a little theory from the school course (briefly):
The values of the lengths of the sides of the triangle are proportional to the sines of the opposite angles:
It generalizes the Pythagorean theorem to arbitrary triangles, so the Pythagorean theorem - becomes a special case of the theorem of cosines.
So, for any triangle, we have the relationship:
After the conversion, we can find the cosine of each angle of the triangle:
and set the following:
If the angle is a direct (second case), the cosine theorem is transformed into Pythagorean theorem.
After considerable layouts and transformation proved "Heron's formula"On which knowing only the sides of the triangle, We can calculate the area:
The above relationships and calculations are used where required calculation of any elements with of considerable size, which can not be measured goniometric ruler or brings a lot of inconvenience the use of roulette.
Examples of problems that can be solved using such theorems
Knowing the length of the slope and angle of the roof, we can get the rest of the values of all the constituent elements, whether it be to the height of the roof ridge or the length of the building:
Conversely, knowing the angle of inclination of the roof and the length of the building with a roof overhang - it is calculated in a couple of actions as the length of rafters and roof height:
But the exact height of the house? - Sure, not a problem!
? / Sin40 ° = 10 / Sin50 °
? = (10 x Sin40 °) / Sin50 ° = 10 x 0.643: 0.766 = 8.4 m.
Determination of the angle of slope
Determine the angle of slope with up to 1 degree from the ground, too, absolutely not straining, you can do: it requires to take the position of the observer, so that the ramp plane fell in line with the line of sight direction.
Now, knowing the height of the house (a) and distance (b), and therefore by the theorem of Pythagoras and the hypotenuse (c), we can calculate the value of the sine or cosine of the angle A (formula in the figure above).
The table below Bradis to help! ))) Find the value in column Sine and compared with the corresponding angle!
The same problems are solved with pediments device in ramp main roof (picture below)! Knowing only the angle of the main roof, we can calculate the length of the rafters and the base crashed into the gable, the angles are equal to each other!
For the construction of buildings and various structures using these formulas are calculated the difference in altitude, and as angles in different planes by means of geodetic devices operating on the basis of the same trigonometry - theodolite, total station and trigonometric leveling.
And this is only a small part of the examples where we need knowledge of trigonometry ...
It seems the teachers were right when they said that "Mathematics is useful !!!" ))).
That's all, thank you for your attention!
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