How to find the angle of a triangle?

There are several ways to calculate the sides and angles of a triangle. These depend on the type of triangle you are working with.

In this opportunity, it will be shown how to calculate the sides and angles of a right triangle, assuming that certain data of the triangle with known.

The elements that will be used are:

- The Pythagorean theorem

Given a right triangle with legs "a", "b" and hypotenuse "c", it is true that "c² = a² + b²".

- Area of ​​a triangle

The formula to calculate the area of ​​any triangle is A = (b × h) / 2, where “b” is the length of the base and “h” is the length of the height.

- Angles of a triangle

The sum of the three interior angles of a triangle is 180º.

- Trigonometric functions:

Consider a right triangle. Then, the trigonometric functions sine, cosine and tangent of the angle beta (β) are defined as follows:

sin (β) = CO / Hip, cos (β) = CA / Hip and tan (β) = CO / CA.

How to find the sides and angles of a right triangle?

Given a right triangle ABC, the following situations can occur:

1- The two legs are known

If leg “a” measures 3 cm and leg “b” measures 4 cm, then the Pythagorean theorem is used to calculate the value of “c”. By substituting the values ​​of "a" and "b" we obtain that c² = 25 cm², which implies that c = 5 cm.

Now, if angle β is opposite leg “b”, then sin (β) = 4/5. By applying the inverse function of the sine, in this last equality we obtain that β = 53.13º. Two internal angles of the triangle are already known.

Let θ be the angle that remains to be known, then 90º + 53.13º + θ = 180º, from which we obtain that θ = 36.87º.

In this case it is not necessary that the known sides are the two legs, the important thing is to know the value of any two sides.

2- A leg is known and the area

Let a = 3 cm be the known leg and A = 9 cm² the area of ​​the triangle.

In a right triangle, one leg can be considered as the base and the other as the height (since they are perpendicular).

Suppose that “a” is the base, therefore 9 = (3 × h) / 2, from which we obtain that the other leg is 6 cm. To calculate the hypotenuse, proceed as in the previous case, and we obtain that c = √45 cm.

Now, if the angle β is opposite the leg “a”, then sin (β) = 3 / √45. Solving for β it is obtained that its value is 26.57º. We only need to know the value of the third angle θ.

It is satisfied that 90º + 26.57º + θ = 180º, from which it is concluded that θ = 63.43º.

3- An angle and a leg are known

Let β = 45º be the known angle and a = 3 cm the known leg, where leg “a” is opposite angle β. Using the tangent formula, we obtain that tg (45º) = 3 / CA, from which it follows that CA = 3 cm.

Using the Pythagorean theorem we obtain that c² = 18 cm², that is, c = 3√2 cm.

It is known that an angle measures 90º and that β measures 45º, from here it is concluded that the third angle measures 45º.

In this case, the known side does not have to be a leg, it can be any of the three sides of the triangle.

References

1. Landaverde, F. d. (1997). Geometry (Reprint ed.). Progress.
2. Leake, D. (2006). Triangles (illustrated ed.). Heinemann-Raintree.
3. Pérez, C. D. (2006). Precalculation. Pearson Education.
4. Ruiz, Á., & Barrantes, H. (2006). Geometries. CR technology.
5. Sullivan, M. (1997). Precalculation. Pearson Education.
6. Sullivan, M. (1997). Trigonometry and Analytical Geometry. Pearson Education.