The value of tan 90° is undefined (not defined). This is because tan θ = sin θ / cos θ, and at 90°, sin 90° = 1 and cos 90° = 0. Since division by zero is undefined in mathematics, tan 90° = 1/0 = undefined. As the angle approaches 90° from below, tan approaches positive infinity (+∞), and from above, it approaches negative infinity (−∞).
tan 90° is undefined (not defined) because tan = sin/cos = 1/0, which is division by zero.
sin 90° = 1 and cos 90° = 0, making tan 90° = 1/0 = undefined.
As angle approaches 90° from below, tan → +∞. From above, tan → −∞.
cot 90° = 0 (cos 90° / sin 90° = 0/1 = 0).
sec 90° is also undefined (1/cos 90° = 1/0 = undefined).
On the unit circle, 90° corresponds to the point (0, 1) — the x-coordinate (cos) is 0.
In radians, tan(π/2) is undefined.
The definition of tangent is: tan θ = sin θ / cos θ = opposite / adjacent
At θ = 90°:
So tan 90° = sin 90° / cos 90° = 1 / 0
Division by zero is undefined in mathematics — you cannot divide any number by 0. Therefore, tan 90° is NOT DEFINED (undefined).
This is written as:
Important distinction: saying tan 90° = ∞ is informal notation. Strictly speaking, tan 90° does not have a numerical value — it is undefined.
As θ approaches 90°, what happens to tan θ?
From the left (approaching 90° from below):
| θ | tan θ |
|---|---|
| 80° | 5.671 |
| 85° | 11.430 |
| 88° | 28.636 |
| 89° | 57.290 |
| 89.9° | 572.96 |
| 89.99° | 5729.6 |
| 90° | undefined |
As θ → 90° from below, tan θ → +∞ (positive infinity)
From the right (approaching 90° from above):
| θ | tan θ |
|---|---|
| 91° | −57.290 |
| 90.1° | −572.96 |
| 90.01° | −5729.6 |
As θ → 90° from above, tan θ → −∞ (negative infinity)
Because the left-hand limit (+∞) and right-hand limit (−∞) are different, the limit does not exist at θ = 90°.
Complete trigonometric table at 90°:
| Function | Value at 90° |
|---|---|
| sin 90° | 1 |
| cos 90° | 0 |
| tan 90° | Undefined |
| cot 90° | 0 |
| sec 90° | Undefined |
| cosec 90° | 1 |
Explanations:
So both tan 90° and sec 90° are undefined.
On the unit circle (radius = 1, centre at origin), any angle θ corresponds to a point: (cos θ, sin θ)
At θ = 90°:
tan θ = y-coordinate / x-coordinate = sin θ / cos θ
At 90°: tan 90° = 1 / 0 = undefined
Geometric interpretation using right triangles: In a right triangle, tan θ = opposite/adjacent. As θ → 90°, the adjacent side → 0 and the opposite side → hypotenuse. The ratio opposite/adjacent → ∞ because we are dividing by an increasingly small number.
At exactly 90°, the adjacent side has length 0, meaning the triangle degenerates (collapses), so the ratio is undefined.
In radians: 90° = π/2, so tan(π/2) is undefined.
| Angle | sin | cos | tan | cot | sec | cosec |
|---|---|---|---|---|---|---|
| 0° | 0 | 1 | 0 | Undefined | 1 | Undefined |
| 30° | 1/2 | √3/2 | 1/√3 | √3 | 2/√3 | 2 |
| 45° | 1/√2 | 1/√2 | 1 | 1 | √2 | √2 |
| 60° | √3/2 | 1/2 | √3 | 1/√3 | 2 | 2/√3 |
| 90° | 1 | 0 | Undefined | 0 | Undefined | 1 |
Memory trick for sin values (0° to 90°): sin 0°=√0/2=0, sin 30°=√1/2=1/2, sin 45°=√2/2=1/√2, sin 60°=√3/2, sin 90°=√4/2=1
Cos values are the reverse: cos 0°=1, cos 30°=√3/2, ..., cos 90°=0.
tan 90° is specifically undefined because the formula tan=sin/cos gives 1/0 at this angle.
tan 90° is undefined (not defined). This is because tan 90° = sin 90° / cos 90° = 1 / 0, and division by zero is undefined in mathematics.
tan 90° is not defined because tan θ = sin θ / cos θ. At 90°, sin 90° = 1 and cos 90° = 0. So tan 90° = 1/0, which is undefined because you cannot divide by zero.
cot 90° = 0. This is because cot θ = cos θ / sin θ = 0 / 1 = 0. Unlike tan 90°, cot 90° is defined and equals zero.
As the angle approaches 90° from below (e.g., 89°, 89.9°, 89.99°), tan increases toward +∞. As it approaches from above (e.g., 91°), tan comes from −∞. At exactly 90°, tan is undefined.
90° = π/2 radians. So tan(π/2) is undefined, same as tan 90°. The function tan(x) has vertical asymptotes at x = π/2 + nπ for any integer n.
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