9 junio, 2024

How to get the angle of a triangle? (Example)

There are various 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 are 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 180°.

– The 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 calculate 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 the leg «a» measures 3 cm and the leg «b» measures 4 cm, then to calculate the value of «c» the Pythagorean theorem is used. By substituting the values ​​of “a” and “b” it is obtained that c²=25 cm², which implies that c=5 cm.

Now, if angle β is opposite to leg “b”, then sin(β)=4/5. When applying the inverse function of the sine, in this last equality it is obtained 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=180°, from which it is obtained 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 it is obtained that the other leg measures 6 cm. To calculate the hypotenuse, we proceed as in the previous case, and it is obtained that c=√45 cm.

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

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

3- An angle and a leg are known

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

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

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

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

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