What fraction is tan 90?
The Correct Answer and Explanation is :
The value of (\tan(90^\circ)) is undefined. This can be explained by understanding the definition of the tangent function and its behavior at this angle.
Definition of Tangent:
The tangent function in trigonometry is defined as the ratio of the opposite side to the adjacent side in a right triangle. More formally, for an angle (\theta) in a right triangle, we have:
[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
]
Alternatively, in terms of sine and cosine, the tangent of an angle can be written as:
[
\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}
]
This formula tells us that the tangent is the ratio of the sine to the cosine of the angle.
Behavior at (90^\circ):
At (\theta = 90^\circ), we know that:
- (\sin(90^\circ) = 1)
- (\cos(90^\circ) = 0)
Thus, the tangent of (90^\circ) is:
[
\tan(90^\circ) = \frac{\sin(90^\circ)}{\cos(90^\circ)} = \frac{1}{0}
]
Since division by zero is undefined in mathematics, (\tan(90^\circ)) is undefined.
Graphical Interpretation:
If you look at the graph of the tangent function, it is a periodic curve that repeats every (180^\circ). The graph approaches infinity as it gets closer to (90^\circ) from the left and negative infinity as it approaches from the right. There is a vertical asymptote at (90^\circ), meaning the function does not have a finite value at this angle, confirming that (\tan(90^\circ)) is undefined.
Conclusion:
In summary, (\tan(90^\circ)) is undefined because the cosine of (90^\circ) is zero, and dividing by zero is not allowed in mathematics. Therefore, there is no numerical value for (\tan(90^\circ)), and it is considered to be undefined.