39. Solving triangles

The resolution of triangles is a fundamental topic in mathematics and can be a decisive point to guarantee a good grade in the Enem. In this chapter, we will cover concepts and practices for solving problems involving triangles.

First of all, it is important to remember that a triangle is a geometric figure composed of three sides and three angles. The sum of the interior angles of a triangle always adds up to 180 degrees. To solve a triangle, you need to know at least three elements, at least one of which is a side.

There are several ways to solve a triangle, depending on the known elements. The most common forms are: by sine, by cosine, by tangent, by the Law of Sines and the Law of Cosines.

Sine, Cosine and Tangent Resolution

If we know an angle and the length of one side, we can use the sine, cosine, and tangent trigonometric relationships to find the other elements of the triangle. The sine of an angle is the ratio of the opposite side to the hypotenuse; the cosine is the ratio of the adjacent side to the hypotenuse; and the tangent is the ratio of the opposite side to the adjacent one.

Law of Sines

The Law of Sines is a formula that relates the sides of a triangle to the sines of its angles. It is particularly useful when we know two angles and a side, or two sides and an angle not included between them. The Law of Sines states that the ratio of the length of one side to the sine of its opposite angle is constant for all three sides of a triangle.

Law of Cosines

The Law of Cosines is another formula that can be used to solve triangles. It is useful when we know the three sides of a triangle, or two sides and the angle between them. The Law of Cosines states that the square of the length of one side is equal to the sum of the squares of the lengths of the other two sides, minus twice the product of the lengths of those sides and the cosine of the angle between them.

Practicing solving triangles is an excellent way to become familiar with trigonometric relationships and the laws of sines and cosines. Furthermore, it is an essential skill for solving many geometry and physics problems.

Finally, it is important to note that solving triangles is just one part of studying mathematics for the Enem. Other topics such as algebra, statistics, spatial geometry and combinatorics are also fundamental. Therefore, it is essential to have a comprehensive and balanced study to ensure good preparation for the test.

We hope this chapter has helped clarify the resolution of triangles. Remember, practice is the key to mastery, so keep working on problems and exercises to hone your skills. Good luck with your studies!

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