﻿ Resolution of right triangles. Trigonometric relations. Solved exercises

Resolution of right triangles. Trigonometric relations. Solved exercises

The resolution of right triangles consists in calculating the measurements of its three sides and the value of its three angles, when we already know at least two of these elements.

Right triangles: What elements do they have? How are the elements of a right triangle related?

Triangles in general are made up of 3 sides and 3 angles. Also, right triangles are called right triangles because they have a right angle between their legs.

The sides of a right triangle are the hypotenuse and the two legs:

The side in front of the right angle is the hypotenuse:

The other two sides are the legs: the major leg and the minor leg, which as their name indicates, the major leg is the one with the longest length and the minor leg is the one with the shortest length.

But there is another way of naming the catetos, depending on the angle we take as reference: the opposite leg and contiguous leg.

How to identify the legs in a right triangle?

I’ll explain how to differentiate between the opposite leg and the contiguous leg according to the reference angle.

How do you know which is the opposite leg?

The side opposite the reference angle is called the opposite leg.

How do you know which is the contiguous cathet or adjacent leg?

The side that is touching that angle is called the contiguous leg.

For example, in this triangle:

If we take the angle B as a reference:

b is the side in front of B, so that is de opposite leg and c is the side touching angle B, so that is the contiguous leg.

But if we take angle C as a reference:

Then b is the contiguous leg and c is the opposite leg.

So, to know which of all the trigonometric relations in the right triangle you have to use to solve a problem, the first thing you have to do is to identify your legs with respect to the angle you are calculating them.

The sides and angles of the right triangle have a series of relations between them, which will help us calculate the measurements of the elements that we do not know.

• The three sides are related by the Pythagorean theorem:

• The three angles sum to each other by 180º:

• The sides and angles are related by trigonometric relations, which I explain in detail in the following section.

Trigonometric relations of a right triangle

The angles and sides of a right triangle are related by some expressions that we call trigonometric relations.

We are going to see them one by one, taking angle B as the reference angle.

Angle B sine

The sine relates angle B to the opposite leg and the hypotenuse. In other words, it is the ratio between the opposite leg and the hypotenuse. It is expressed as sen B:

Angle B cosine

It relates the angle B with the contiguous leg and the hypotenuse. It is the ratio between the contiguous len and the hypotenuse. It is expressed as cos B:

Angle B tangent

It is the relation between the opposite leg and the contiguous leg. Also between the sine and the cosine. It is expressed as tg B:

Angle B cosecant

It is the inverse relation of the sine. It is expressed as cosec B:

It should not be confused with the inverse function of the sine, which is the arc sine.

Angle B secant

It is the inverse relation of the cosine. It is expressed as cosec B:

Not to be confused with the inverse function of the cosine which is the cosine arc.

Angle B contangent

It is the inverse ratio of the tangent and is expressed as cotg B:

It should not be confused with the inverse function of the tangent which is the tangent arc.

As you can see, all trigonometric ratios associate an angle with two sides, that is, three variables. Therefore, when choosing which reason to use, it must be the one that we know at least two of the three variables.

We have to play with these formulas according to the data given by the statement of the problem.

There are two possible cases that we can find in the problems or exercises of resolution of right triangles that are:

• That we know two sides and ask us for some angle or the other side
• That we know a side and one angle and we are asked to calculate any other side or angle

If you give us two angles as data, we will not be able to calculate the sides of that right triangle. We would need more information.

We always need at least two data to calculate a third.

Resolution of right triangles when two sides are known

Let’s see, everything I just explained with an example. We have a triangle of which we know 2 of its sides:

We are asked to calculate side b and angle B:

To calculate side b, we do so using the Pythagoras formula, since in that formula the 3 sides are related and only 1 side remains to be known.

From Pythagoras’ formula we clear the major leg, which corresponds to side b:

And now we substitute values and calculate:

• C = b (this is the side we are calculating)
• c = 3 m
• H = 5 m

We are left with an equation, of which we have b as an unknown.

To calculate the angle B, we can do it in many ways. One of them is using the sine trigonometric ratio for example, since we know the value of the hypotenuse and the opposite leg.

We could actually use any trigonometric reason because we know all its sides:

We substitute values and solve:

Once we know the sine value of B, with the calculator we have calculated its inverse and we obtain the angle. Pay attention that the calculator is in degrees and not in radians.

Resolution of right triangles when one side and one angle are known

Let’s see another example with this triangle, of which we know an angle and a side.

Nos piden calcular el lado c:

The Pythagoras formula cannot be used because I only have the data of one side. Therefore, it remains to use the trigonometric ratios.

We know the hypotenuse and they are asking us for the adjacent side. The ratio that relates these two sides is the cosine:

We substitute the values we know and solve:

• Angle B = 60º
• Hypotenuse = 6 m

In this case, the 60 c0sene, once solved with the calculator, we treat it as a number.

Now that we know c, we could use Pythagoras to calculate the side we have left, if they were asking for it.

As we get to know more elements, we have the possibility of applying more relations to find the solution they ask us for.

The resolution of right triangles has many applications, such as calculating the distance of a cable, calculating the height of a tower, calculating the distance of a tree according to its shadow, calculating the angle formed by a ladder resting on a wall or anything else that the statement of a problem may ask us to do.

Resolved exercise of resolution of right triangles

In the following exercise, we will apply what we learned to calculate different angles and sides of a triangle, which in principle is not a rectangle.

Calculate the sides and angles of the next triangle, from which we get one of the angles, one of the sides and the distance from point A to point D:

The first thing you have to do is look for right triangles in this triangle in order to solve it and calculate the missing elements.

To do this, we draw a vertical line that joins vertex B with point D, which is the height of the triangle and divides it into two right triangles:

We start by keeping the triangle on the left:

In this triangle we need to calculate side c and side h (I have called them c and h for calling them in some way to distinguish them).

Let’s calculate side c. We know the angle and the side adjacent to the angle, but we don’t know the opposite side.

The ratio that relates these three variables is the tange:

From where we clear c:

We calculate the 45º tangent with the calculator and finally we operate:

The c side measures 4 cm.

Now we will calculate side h, by means of the sine ratio, since we know the angle, the opposite leg and we do not know the hypotenuse:

We clear h:

We calculate the 45º sine with the calculator and we operate:

The h side measures 5.65 cm.

Another way of calculating h would have been by the Pythagorean theorem.

Finally we are going to calculate the missing angle. As we know that the sides of a triangle must add 180º, from 180º we subtract the two angles we already know:

So the missing angle measures 45º.

We place all the calculated data in the right triangle:

We do the same with the right triangle on the right, where we place the value of one of the sides that we have calculated thanks to the other right triangle:

Of this triangle would be missing to calculate the side C and the two angles that are not right.

We begin by calculating the alpha angle. I will do it by means of the sine reason, which relates the opposite leg to the hypotenuse and I have the two data:

I calculate the sine of alpha

And through the inverse of the sine, I calculate the angle:

The alpha angle measures 19.26º.

Side C is calculated by Pythagoras this time:

Side C measures 11.31 cm.

We will calculate the missing angle knowing that the three angles of a triangle add 180º and 180º I will subtract the two angles we already know:

I place all the data in the triangle:

Now I have all the data I need to calculate the sides and angles of the original triangle.

To calculate the angle B, I add the two angles I have obtained previously:

And to calculate the lower side, I add the data I already had from point A to point D and the one I calculated from point D to point C:

So the sides and angles of the original triangle are: