Sin Cos Tan Calculator
Enter an angle to calculate all six trigonometric functions instantly. Switch between degrees and radians at any time.
The Six Trigonometric Functions
For a right triangle with an acute angle $\theta$, opposite side $o$, adjacent side $a$, and hypotenuse $h$:
| Function | Definition (triangle) | Unit circle | Reciprocal of |
|---|---|---|---|
| sin θ | $o / h$ | $y$-coordinate | — |
| cos θ | $a / h$ | $x$-coordinate | — |
| tan θ | $o / a$ | $y / x$ | — |
| csc θ | $h / o$ | $1 / y$ | sin θ |
| sec θ | $h / a$ | $1 / x$ | cos θ |
| cot θ | $a / o$ | $x / y$ | tan θ |
When Functions Are Undefined
Some functions are undefined for certain angles because they would require dividing by zero:
- tan θ and sec θ are undefined at $90° + 180°n$ (i.e. …, −90°, 90°, 270°, …) because $\cos\theta = 0$ at those points.
- csc θ and cot θ are undefined at $0°, 180°, 360°, \ldots$ (multiples of 180°) because $\sin\theta = 0$ there.
Key Identities to Remember
The most fundamental identity connects sin and cos:
$$\sin^2\theta + \cos^2\theta = 1$$Dividing through by $\cos^2\theta$ or $\sin^2\theta$ gives two more:
$$\tan^2\theta + 1 = \sec^2\theta \qquad 1 + \cot^2\theta = \csc^2\theta$$Use the degrees to radians converter if you need to convert your angle first, or explore the unit circle calculator to see where these values come from geometrically.
Frequently Asked Questions
Why is tan(90°) undefined?
$\tan\theta = \sin\theta / \cos\theta$. At 90°, $\cos(90°) = 0$, so we'd be dividing by zero — the result is undefined (approaches ±∞ as you approach 90° from either side).
Do trig functions work for angles greater than 90°?
Yes. The unit circle definition extends all trig functions to any angle. Signs change by quadrant: sin is positive in Q1 and Q2; cos in Q1 and Q4; tan in Q1 and Q3. The phrase "All Students Take Calculus" (ASTC) is a common mnemonic for which functions are positive in each quadrant.
What's the difference between sin⁻¹ and csc?
$\csc\theta = 1/\sin\theta$ is the reciprocal function. $\sin^{-1}(x)$ (arcsin) is the inverse function — it takes a ratio and returns an angle. They are not the same. See the inverse trig calculator for arcsin, arccos, and arctan.
