Understanding the Quotient Identity for Tangent: Exploring the Relationship between Sine and Cosine in Trigonometry

Quotient Identity: tan(x) =

The quotient identity for tangent is written as tan(x) = sin(x) / cos(x)

The quotient identity for tangent is written as tan(x) = sin(x) / cos(x).

In trigonometry, the tangent function relates the ratio of the opposite side to the adjacent side of a right triangle. It is defined as the sine of an angle divided by the cosine of the same angle.

To understand how this identity is derived, let’s consider a right triangle with an angle x. In this triangle, the side opposite to the angle x is denoted by “opp” and the side adjacent to the angle x is denoted by “adj”.

According to the definitions of sine and cosine, we have:

sin(x) = opp/hypotenuse
cos(x) = adj/hypotenuse

Now, we can express the tangent function as the ratio of sin(x) to cos(x):

tan(x) = sin(x) / cos(x)
= (opp/hypotenuse) / (adj/hypotenuse)
= opp/adj

Simplifying further, we can cancel out the common hypotenuse terms:

tan(x) = opp/adj

So, the quotient identity for tangent is tan(x) = sin(x) / cos(x), which can also be written as tan(x) = opp/adj.

This identity is used in various trigonometric calculations, such as finding angles or side lengths in right triangles or solving trigonometric equations.

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