The tangent function (tan) is commonly used in trigonometry to find the ratio between the length of the side opposite to an angle in a right triangle and the length of the adjacent side. However, when the angle is 90 degrees, the tangent function becomes undefined.
To understand why, let's consider the definition of the tangent function:
tan(theta) = opposite/adjacent
In a right triangle, the tangent of an angle theta is the ratio of the length of the side opposite to the angle to the length of the side adjacent to the angle.
When the angle is 90 degrees, the tangent becomes:
tan(90 degrees) = opposite/adjacent
Now, in a right triangle with a 90-degree angle, we have a vertical side (opposite) and a horizontal side (adjacent), and the tangent is defined as the ratio of the vertical side to the horizontal side. However, when the angle is 90 degrees, the horizontal side has a length of zero. In other words, the horizontal side does not exist in this case.
So, dividing the vertical side (opposite) by a non-existent horizontal side (adjacent = 0) leads to an undefined value:
tan(90 degrees) = opposite/0
Any division by zero leads to undefined results in mathematics. Therefore, tan(90 degrees) is undefined.
Mathematically, we say that the limit of the tangent function as the angle approaches 90 degrees is undefined:
lim(theta->90 degrees) tan(theta) = undefined
That's why the tangent of 90 degrees is considered undefined in trigonometry.
Why is Tan 90 Undefined. Explain it in detail along with mathematical expressions and equations.
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