What is the tan of 90?
Understand the Problem
The question is asking for the value of the tangent function at 90 degrees. In trigonometry, the tangent of an angle is defined as the ratio of the opposite side to the adjacent side in a right triangle. At 90 degrees, the tangent function approaches infinity because the adjacent side length becomes zero.
Answer
The value of $\tan(90^\circ)$ is undefined.
Answer for screen readers
The value of the tangent function at 90 degrees is undefined.
Steps to Solve
-
Identifying the angle
We are given the angle of 90 degrees. In trigonometric terms, this angle is also expressed in radians as $\frac{\pi}{2}$. -
Understanding the tangent function
The tangent function is defined as the ratio of the sine and cosine of an angle:
$$ \tan(x) = \frac{\sin(x)}{\cos(x)} $$ -
Calculate sine and cosine at 90 degrees
At 90 degrees, we know:
$$ \sin(90^\circ) = 1 $$
$$ \cos(90^\circ) = 0 $$ -
Applying the tangent definition
Using the values from the previous step:
$$ \tan(90^\circ) = \frac{\sin(90^\circ)}{\cos(90^\circ)} = \frac{1}{0} $$ -
Value of tangent at 90 degrees
Dividing by zero is undefined in mathematics, which means:
$$ \tan(90^\circ) = \text{undefined} $$
The value of the tangent function at 90 degrees is undefined.
More Information
The tangent function approaches infinity as the angle approaches 90 degrees from the left side. This makes sense because as the angle gets closer to 90 degrees, the adjacent side of the triangle gets closer to zero, making the ratio infinitely large.
Tips
- A common mistake is to assume that $\tan(90^\circ)$ has a specific numerical value instead of recognizing it as undefined. To avoid this mistake, always remember that the tangent function has discontinuities at specific angles like 90 degrees.
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