TYPES of TRIGONOMETRIC identities

From unProfesor we are pleased to publish a lesson on the types of trigonometric identities. In this lesson you will be able to understand what trigonometric identities are and what types there are. To finish, you can do some training, of which we leave you their respective solutions so that you can make sure that you have understood what is explained in the article.

The trigonometry is that branch of mathematics, specifically geometry, which focuses on the relationship between the sides and angles of triangles. In this way, it takes care of the functions associated with angles, which are known as trigonometric or circular functions: sine, cosine, tangent, secant...

The trigonometric identities, which are the ones we are going to study in this lesson, are those equalities that contain trigonometric functions, so they can be of different types, as we will see later. continuation.

The trigonometric identities can be classified in a particular way. For your better understanding, here is a summary of the different types of trigonometric identities.

1. reciprocal identities

They are formed by the product of two reciprocal ratios.

• Sine = 1 / Cosecant
• Cosine = 1 / Secant
• Tangent = 1 / Cotangent

2. Quotient Identities

They are formed by division.

• Tangent = Sine / Cosine
• Cotangent = Cosine / Sine

3. Pythagorean identities

The Pythagoreans are another type of trigonometric identities. They are formed by applying the Pythagoras theorem.

• Breast² + Cosine² = 1
• Drying² = Tangent² + 1
• Cosecant² = Cotangent² + 1

To demonstrate the different types of trigonometric identities that we have mentioned, we must develop them as in the following example, which will help you to solve the activities that we will propose later:

Cotangent Secant = Cosecant

• We start by using the cotangent and secant identities, which are cosine / sine and 1 / cosine, respectively.
• We have taken the first one directly from the second identity by quotient, while we have taken the second one by isolating the reciprocal second identity. That is, if cosine = 1 / secant, isolating we obtain that secant = 1 / cosine.
• Once we have this, we continue with the equality, like this: Cotangent · Secant = (cosine / sine) * (1 / cosine).
• We operate: Cotangent · Secant = Cosine / (Sine * Cosine).
• Since the cosine is in both the numerator and the denominator, we can eliminate it and we are left with Cotangent · Secant = 1 / Sine.
• We know from the first reciprocal formula that sine = 1 / cosecant, so if we isolate, we know cosecant = 1 / sine.
• Thus, since our result was 1 / sine, it will also be cosecant, since it is an equality.
• Finally, we can conclude that Cotangent · Secant = Cosecant.

The conclusion is that, in order to prove an identity or simplify trigonometric expressions, we will have to remember of which are the trigonometric identities and go making the pertinent substitutions, until arriving at the expression desired.

To test what you have learned reading this lesson, we suggest you do the following exercise, taking as a reference the procedure explained in the example above:

1. Check the following identity: Sine Secant = Tangent

We are going to see the answer to the activity proposed in the previous section, in order to check that you have understood what has been explained throughout this article:

1.

• Sine Secant = Tangent
• Since we know that secant = 1 / cosine, which we get from isolating the second reciprocal identity, Well, we write the statement again, but where it says secant we will put 1 / cosine: sine * (1 / cosine).
• We operate and we are left with sine / cosine. If we go to the first identity by quotient, we know that tangent = sine / cosine, so the result we had was the same as the tangent.

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