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Sine theorem and cosine theorem

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The cosine theorem can be applied to either side of a triangle.

We write down the formulas for each side and find out how to apply the cosine theorem depending on the conditions of the problem.

The square of either side of the triangle is the sum of the squares of the other two sides minus the double product of these sides by the cosine of the angle between them.

For triangle ABC

can be recorded

in one of three variations:

we obtain the following three formulas of the cosine theorem:

To which side of the triangle is the cosine theorem applied?

The cosine theorem is applied to the side opposite which the angle is determined (that is, it is either known, or it just needs to be found).

Next, we consider the application of the cosine theorem in solving problems.

The formula of the cosine theorem

The squared side of the triangle is the sum of the squares of the other two sides minus the double product of these sides by the cosine of the angle between them.

That is, for a flat triangle with sides $ a $, $ b $ and $ c $ and angle $ alpha $, opposite the side $ a $, the following relation holds:

The cosine theorem is a generalization of the Pythagorean theorem. Statements generalizing the Pythagorean theorem and equivalent to the cosine theorem were formulated separately for cases of an acute and obtuse angle in sentences 12 and 13 of the second book of the Beginning of the ancient Greek mathematician Euclid (c. 300 BC). Assertions equivalent to the cosine theorem for a spherical triangle were used in the writings of mathematicians in Central Asia. The cosine theorem for the spherical triangle in the usual form was formulated by the outstanding German astrologer, astronomer and mathematician Regiomontan (1436 - 1476), calling it the “Albathegnia theorem” (named after the outstanding medieval astronomer and mathematician Abu Abdallah Muhammad ibn Jabir ibn Sinan al-Batani (85 - 929).

In Europe, the cosine theorem was popularized by the French mathematician Francois Viet (1540 - 1603) in the 16th century. At the beginning of the 19th century, it began to be written down in the algebraic notation accepted to this day.

Corollary to the cosine theorem

The cosine theorem can be used to find the cosine of the angle of a triangle:

If $ b ^ <2> + c ^ <2> -a ^ <2>> 0 $, then the angle $ alpha $ is acute,

If $ b ^ <2> + c ^ <2> -a ^ <2> = 0 $, then the angle $ alpha $ is a straight line,

The task. In the triangle $ ABC AC = 3, BC = 5 $ and $ AB = 6. $. Find the angle opposite to the side $ AB $

Decision. According to the corollary of the cosine theorem, we have:

$$ angle A C B = arccos left (- frac <1> <15> right) $$

Answer. $ angle A C B = arccos left (- frac <1> <15> right) $

The task. Given a triangle is $ ABC $, whose side lengths are $ AC = 17, BC = 14, angle ACB = 60 ^ < circ> $. Find the length of the third side of the triangle in question.

Decision. According to the cosine theorem

$$ A B ^ <2> = A C ^ <2> + B C ^ <2> -2 cdot A C cdot B C cdot cos angle A C B = $$

$$ = 17 ^ <2> + 14 ^ <2> -2 cdot 17 cdot 14 cdot cos 60 ^ < circ> = 289 + 196-238 = 24 $$

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