4.09.2016

The equation a fraction

What if the equation of lead shot, and X is hidden in the denominator? We need to get rid of the denominator. What will happen with X? Do not worry. If you do everything correctly, he from the denominator goes into the numerator. Here you can recommend two ways to solve the equation.

The first method is to bring the expression to normal proportions. Then we use the proportion of the property - the multiplication of extreme terms in proportion equal to the multiplication of the average proportion of members. Denominator magically disappear. We get a simple equation, which can be easily solved. Consider an example.

The equation a fraction. Mathematics For Blondes.
The equation a fraction

The number did not work nicely. Make checks and look at the result. Substitute the original expression instead of X its value.


Checking equation. The equation a fraction. Mathematics For Blondes.
Checking equation

The audit showed that the equation is solved correctly.

The second method is to bring the expression to a common denominator. The denominator should be common to the left and right sides of the equation. In this case, we in good conscience can throw the denominator in the trash. We see an example of reduction to a common denominator.

The equation a fraction. Method is to bring the expression to a common denominator. Mathematics For Blondes.
The equation a fraction
The result was the same as in the first case.

4.01.2016

Law of cosines in general

To present the law of cosines in general need to go back to the beginning. If the expression for one side of the triangle the length multiplied by the same hand n-times, the equation does not change.

Conversion formulas. Law of cosines in general. Mathematics For Blondes.
Conversion formulas

If you add equal to three sides of the triangle, we obtain the law of cosines in general.

Law of cosines in general. Mathematics For Blondes.
Law of cosines in general


The law of cosines in general terms describes the relationship between the sides and angles of a triangle in a multidimensional space. If the given equation is satisfied, then the triangle is in the Euclidean space. Options description triangle in curvilinear spaces require further study. For the correct application of the law of cosines in solid geometry, spherical triangles side length should be unit of measurement in the same units of measurement, which is measured in planimetrics.

3.28.2016

Degenerate triangle

Finally, we will check the law of cosines to the perimeter in the degenerate triangle. There can be two options. If we decrease the base of an isosceles triangle to zero, we obtain the two overlapping segments. The sum of the angles of this degenerate triangle is 180 degrees.

Degenerate-triangle. Law of cosines to the perimeter. Mathematics For Blondes.
Degenerate triangle

If we combine the upper vertex of the triangle with the base, we get the second type of degenerate triangles. This segment equal to the sum of the other two segments. The sum of the angles are also equal to 180 degrees.

Degenerate triangle. The sum of two segments. Low of cosines. Mathematics For Blondes.
Degenerate triangle

A degenerate triangle - is the lower limit of the application of the law of cosines. After that we will look at the law of cosines in general.

3.24.2016

Isosceles obtuse triangle

Now we will check how the law of cosines in obtuse triangle. For example, consider an isosceles obtuse triangle.

Isosceles obtuse triangle. Law of cosines. Mathematics For Blondes.
Isosceles obtuse triangle


Ups! The perimeter of the triangle may be square roots. The law of cosines is not working? Do not jump to conclusions. The secret is revealed very simple. Let us express the base of the triangle through the sides and see what happens. Double-paste the resulting equation in our result. First time turvy-topsy, second time topsy-turvy.

The perimeter of the triangle. Mathematics For Blondes.
The perimeter of the triangle

The law of cosines to the perimeter works flawlessly. We have completed our review of the degenerate triangle.

P.S. 05/31/16 The онлайн игрыcalculator of triangles confirms the received result.

To calculate a triangle. Mathematics For Blondes.
To calculate a triangle

3.16.2016

Right triangle

Now we will check the law of cosines to the perimeter of the example of a right triangle. At one corner of the right triangle is 90 degrees, the cosine of this angle is zero. Cosine of the other angles are obtained by dividing the length of the hypotenuse to the length of the adjacent side. In general, the law of cosines check looks like.

Right triangle. Law of cosines to the perimeter. Mathematics For Blondes.
Right triangle

Now we substitute in formula for the length of the sides of a right triangle and values of cosines.

Right triangle. Mathematics For Blondes
Right triangle

These values are equal to the perimeter of the triangle. Cosine law allows to check the triangle on the break. That's what happens when one of the sides does not reach the top of the triangle.

Broken triangle. Mathematics For Blondes.
Broken triangle

Equality is performed, but the result is not equal to the sum of the lengths of a broken line or the perimeter of the triangle. You can then proceed to test the law of cosines in the isosceles obtuse triangle.

3.15.2016

Check the law of cosines

Check the law of cosines for the perimeter, we start with an equilateral triangle. All sides of an equilateral triangle are equal. All angles are equal to 60 degrees. The perimeter of the triangle is equal to three times the length of the sides.

Equilateral triangle. Check the law of cosines. Mathematics For Blondes.
Equilateral triangle

Equality is performed. Now we check the law of cosines in a right triangle.

3.14.2016

Law of cosines for perimeter

Carefully look at the proof of the law of cosines and make some corrections.

Analysis of proof. Law of cosines and perimeter. Mathematics For Blondes.
Analysis of proof

Now we can write the law of cosines for the perimeter of the triangle.

Law of cosines for perimeter. Perimeter of triangle. Mathematics For Blondes.
Law of cosines for perimeter

The result is a very simple and beautiful formula that describes the entire triangle. The law of cosines shows a relationship between the angles and the lengths of the sides of a triangle.

The geometry of the law of cosines. Triangle. Mathematics For Blondes.
The geometry of the law of cosines

Law of cosines for perimeter describes the perimeter of the triangle made up of one-dimensional Euclidean spaces. For multidimensional spaces cosine law has a different view.