Understanding the Tangent Function: Exploring the Relationship Between Opposite and Adjacent Sides in Trigonometry

tan θ

The tangent function, usually written as tan θ, is a trigonometric function that relates the ratio of the length of the opposite side to the length of the adjacent side of a right triangle to the measure of one of its acute angles (θ)

The tangent function, usually written as tan θ, is a trigonometric function that relates the ratio of the length of the opposite side to the length of the adjacent side of a right triangle to the measure of one of its acute angles (θ).

Mathematically, we can define the tangent function as:

tan θ = opposite/adjacent

To better understand this, let’s consider a right triangle:

|
|\
| \
| \
| \
| \
| \
|______\
adjacent

Here, the angle θ is formed between the hypotenuse and the adjacent side. The opposite side is perpendicular to the adjacent side.

To find the value of tan θ, we divide the length of the opposite side by the length of the adjacent side.

It’s also important to note that the tangent function can be positive, negative, or undefined, depending on the value of θ and the quadrants in which it lies.

In the first quadrant (0° to 90°), the tangent function is positive because both the opposite and adjacent sides are positive. In the second quadrant (90° to 180°), the tangent function is negative because the opposite side is positive but the adjacent side is negative. In the third quadrant (180° to 270°), the tangent function is positive again because both the opposite and adjacent sides are negative. In the fourth quadrant (270° to 360°), the tangent function is negative because the opposite side is negative but the adjacent side is positive.

For example, if we have a right triangle where the opposite side has a length of 5 units and the adjacent side has a length of 3 units, we can find the value of tan θ as follows:

tan θ = opposite/adjacent
= 5/3

So, in this case, tan θ = 5/3.

Please note that when the adjacent side is equal to zero, the tangent function is undefined because division by zero is undefined. Additionally, the tangent function is periodic with a period of 180°, so tan(θ + 180°) = tan θ.

I hope this explanation helps! If you have any further questions, feel free to ask.

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