Trigonometric ratios. Fundamental equation of trigonometry. Exercises

I am going to explain how to calculate trigonometric ratios from one of them and for that you must know beforehand the formulas that relate the different trigonometric ratios: the fundamental relationship of trigonometry and other frequently used trigonometric ratios.

We’ll see how to apply them step by step with some determined exercises.

Fundamental equation of trigonometry

For any angle, the trigonometric segments that define the sine and cosine form a right triangle with the radius, which has length 1:

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So by applying the Pythagorean theorem in this triangle, we have the sine squared plus the cosine squared equals 1:

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This is known as the fundamental equation of trigonometry and relates the sine and cosine in the same formula.

Trigonometric relations of frequent use

Let’s now look at other formulas that relate to trigonometric ratios to each other.

I’m not going to stop at demonstrating each one of them, because it’s more important for me that you know how to apply them than you know how to demonstrate them. Therefore I recommend that you learn them and keep them in mind to solve your exercises.

The first is this, which relates the tangent, which is the relationship between the sine and the cosine:

relación trigonometricas

We also know that the tangent is the inverse of the cotangent (and vice versa), so they are related by this formula:

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And therefore, the cotangent can also be expressed as a function of the sine and cosine, inverting the formula of the tangent seen above:

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On the other hand, if we divide the fundamental relationship of trigonometry between cosine squared, we have this relationship:

teorema fundamental de la trigonometria

And this one if we divide the fundamental relationship between the squared sine:

relationships between trigonometric ratios

Both the fundamental relationship and these formulas are used to solve trigonometric equations or to calculate the value of trigonometric ratios by knowing the value of others, without using their inverse trigonometric ratios, which is exactly what we will see below.

Depending on the reason we know and the reason we want to calculate, we will have to apply the formula that best suits us, as each of them relates different trigonometric ratios.

We’ll take it one step at a time.

Exercises resolved on trigonometric identities

We are going to solve a couple of exercises on how to calculate the rest of trigonometric ratios, knowing only one of them.

Exercise resolved 1

Knowing that the sen α 1/3 and that α is between 0º and 90º, calculate the rest of the trigonometric ratios of the angle α.

We have to:

fundamental formulas of trigonometry

Let’s start by applying the fundamental relationship, since it relates the cosine and the sine:

sine to cosine ratio

As we already know the value of the sine, we can obtain the value of the cosine, clearing it from this formula. It’s the only relationship where we can clear the cosine. Therefore, we replace the breast with its value:

cosine squared

All that remains now is to operate and clear the cosine. To do this we clear the cosine squarely:

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We’ll take care of the parenthesis:

relación trigonometrica

We operate on the second member:

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We pass the square of the cosine to the second limb as root and operate:

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We have a positive breast value left.

The statement tells us that the angle α is between 0º and 90º. As we see in the lesson on trigonometric ratios of any angle, the angle is in the first quadrant:

use of trigonometric ratios

Where the sine and cosine are positive, then the value obtained is correct.

As we already know the sine and cosine values, we use this formula to calculate the tangent:

reasons trigonometricas inversas ejercicios

We replace the sine and cosine by their values and operate:

calcular senoAnd we have already calculated the three main trigonometric ratios.

Let’s see another exercise.

Exercise resolved 2

Knowing that the tg α=3/4 and that α is between 180º and 270º, calculate the rest of the trigonometric ratios of the angle α.

They give us a tip:

calculate the tangent

In principle, the most immediate thing you can think of is to use the tangent formula:

find trigonometric reasons of an angle

And since the numerator is the sen α and the denominator is the cos α, you might think that:

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how to clear conseno de un angulo

Well, this is not right. The sine and cosine can be a maximum of 1 and a minimum of -1 and in this case they are 3 and 4 respectively, so THIS SOLUTION IS NOT CORRECT.

Then how do we do it?

The fundamental relationship between trigonometry comes into play again:

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Besides the fact that the sine and cosine can be worth between -1 and 1, they must meet the fundamental relationship.

Therefore, matching the formula of the tangent:

fundamental equation of trigonometry

with the data that gives us the statement, we have:

how to calculate the tangent

From here we clear the sine, passing the cosine and multiplying the second member of the equation:

related reasons resolved exercises

And this expression of the breast is replaced in the fundamental relationship:

calculate trigonometric ratios knowing the sine

We are left with a formula that has only the cosine as an unknown factor. So, let’s clear the cosine.

First of all, we solve the parenthesis:

calculate angle from sine

We’re taking a common factor out of the square cosine on the first limb:

breast calculation

We operate within the parentheses:

cosine squared mennos sine squared

And we leave the square cosine only at the first limb. The 16 that is dividing, passes multiplying the second member and the 25 that is multiplying, passes dividing:

relationships between trigonometric ratios

Now we pass the square of the cosine as root to the second member:

relationships between trigonometric ratios

And we solve the root:

calculate sine and cosine

We have a positive cosine value left.

The statement tells us that α is between 180º and 270º. As we see in the lesson on trigonometric ratios of any angle, the angle is in the third quadrant:

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And therefore, the cosine must be negative, so:

how to clear an angle of a breast

In the expression where we clear the breast:

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We replaced the cosine value we just calculated:

sine by cosine

And we operate, getting the value of the breast:

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That it is negative and therefore corresponds to the sign that the breast must have in the third quadrant.