Trigonometry: worked example
| Angle (degrees) | 30 |
|---|---|
| Sine | 0.5 |
| Cosine | 0.866 |
| Tangent | 0.577 |
Trigonometry: formula
Enter an angle in degrees. The calculator converts to radians by multiplying by pi over 180 and computes sine, cosine and tangent. Sine and cosine are always between minus 1 and 1, while tangent can be any value and is undefined at 90 and 270 degrees.
For a right triangle, sine is the opposite side over the hypotenuse, cosine is the adjacent side over the hypotenuse and tangent is opposite over adjacent.
Trigonometry: changing Angle (degrees)
| Angle (degrees) | Sine | Cosine | Tangent |
|---|---|---|---|
| 24 | 0.407 | 0.914 | 0.445 |
| 27 | 0.454 | 0.891 | 0.51 |
| 30 (example) | 0.5 | 0.866 | 0.577 |
| 33 | 0.545 | 0.839 | 0.649 |
| 36 | 0.588 | 0.809 | 0.727 |
Trigonometry: compare input ranges
| Input | Tested from | To | Sine: low input | Sine: high input | Swing |
|---|---|---|---|---|---|
| Angle (degrees) | 27 | 33 | 0.454 | 0.545 | 0.091 (18%) |
Trigonometry: individual input comparisons
| Input | Value | Sine | Cosine | Tangent |
|---|---|---|---|---|
| Angle (degrees) | 24 | 0.407 | 0.914 | 0.445 |
| 27 | 0.454 | 0.891 | 0.51 | |
| 30 | 0.5 | 0.866 | 0.577 | |
| 33 | 0.545 | 0.839 | 0.649 | |
| 36 | 0.588 | 0.809 | 0.727 |
Trigonometry: which assumptions move the result?
Trigonometry: values and definitions
| Quantity and definition | Value used |
|---|---|
| Angle (degrees) | 30 |
| Sine | 0.5 |
| Cosine | 0.866 |
| Tangent | 0.577 |
Trigonometry: half and double Angle (degrees)
| Angle (degrees) | Sine |
|---|---|
| 15 | 0.259 |
| 30 | 0.5 |
| 60 | 0.866 |
Common error: Expecting a tangent at 90 degrees. The ratio has no finite value there because the horizontal run is zero.
Make sure your angle mode matches your problem: a value in radians entered as degrees gives a wrong answer. Tangent blows up near 90 degrees, so results become huge and unreliable. Inverse functions have restricted ranges and give only one of several possible angles.
Rounding errors appear for angles near multiples of 90.
Geometry calculators: Golden Ratio Calculator, Midpoint Calculator, Rectangle Area & Perimeter Calculator, Circle Area & Circumference Calculator, Triangle Area (Heron's Formula) Calculator, Sphere Volume & Surface Area Calculator.
Sine, cosine and tangent are the three ratios of a right triangle's sides, and they depend only on the angle. At 30 degrees the sine is exactly one half, the cosine is about 0.866 and the tangent is about 0.577. The tangent doubles as a slope: a 30 degree ramp rises 0.577 for every unit it runs, which is how builders turn an angle into a rise.
Check the angle unit. A calculator set to radians reads 30 as a very different angle and returns values that look plausible but are wrong.
Solve triangles on the right triangle, law of sines and triangle side calculators, and compute arcs on the arc length and sector page. The Pythagorean theorem page finds sides, and the dot product calculator uses cosines of angles between vectors. The slope calculator relates to the tangent of an angle.
Why is tangent undefined at 90 degrees?
Because the adjacent side becomes zero, and dividing by zero is undefined, so the tangent grows without bound near 90 degrees.
What are sine, cosine and tangent?
Ratios of sides in a right triangle: sine is opposite over hypotenuse, cosine is adjacent over hypotenuse and tangent is opposite over adjacent.
How do I convert degrees to radians?
Multiply by pi over 180. So 180 degrees is pi radians, and 90 degrees is pi over 2.