Why is Tan 90 Undefined. Explain it in detail along with mathematical expressions and equations.
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.
The tangent function is defined as the ratio of the length of the side opposite to an acute angle in a right triangle to the length of the adjacent side. In other words, for a given angle θ, the tangent of θ is defined as:
tan(θ) = opposite side / adjacent side
Now, let's consider the angle 90 degrees. In a right triangle, the side opposite the right angle is called the hypotenuse. Since there is no adjacent side to the right angle, the tangent function cannot be defined for this angle.
To illustrate this, let's assume we have a right triangle with one angle equal to 90 degrees. In this case, one of the sides forming the right angle (the adjacent side) will have a length of zero. Therefore, we would have:
tan(90) = opposite side / 0
Now, when we divide any number by zero, we encounter an undefined result. This is because division by zero violates the fundamental properties of arithmetic. Mathematically, we cannot determine a specific value for tan(90) because it involves a division by zero, which is undefined.
In conclusion, the tangent of 90 degrees is undefined because there is no adjacent side in a right triangle with a right angle, and dividing by zero is mathematically undefined.