Library / Geometry, coordinates, surveying and trigonometry / GEO-49
Geometry, coordinates, surveying and trigonometry
General solution of a tangent equation
All solutions of tan x = u, from the principal value arctan u and any whole number k.
- Ref
- GEO-49
- Rev
- v1
- Field
- Geometry, coordinates, surveying and trigonometry
- Result
- xk [rad]
The equation tan x = u has solutions xₖ = arctan u + kπ for every whole number k, because the tangent repeats every half-turn. The calculator’s arctan returns the principal value between −π/2 and π/2. Pick the values of k that put x inside the interval a problem asks for.
All solutions of tan x = u, from the principal value arctan u and any whole number k. Choose the k values that put x inside the interval a question asks for.
Calculator
- Principal value, arctan u−0.7854 rad
- Solution in degrees315 °
- Solution reduced into 0 to 2π (k even or odd handled)5.498 rad
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- For tan x = −1 and k = 2 the solution is 5.4978 rad (315°); every other solution differs from it by a whole number of π.
- To find the solutions between 0 and 2π take k = 0 and 1 if the principal value is positive, or k = 1 and 2 if it is negative.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (rad) | Local step | Change in result |
|---|---|---|---|---|---|---|
| k Whole number of half-turns | −5 to 5 | −16.49 | 14.92 | 31.42 | +1 | +57.1 % |
| u Value of tan x | −10 to 10 | 4.812 | 7.754 | 2.942 | +0.4 | +4.46 % |
Sweep of Value of tan x
| Value of tan x | Solution (rad) |
|---|---|
| −10 | 4.812 |
| −7.5 | 4.845 |
| −5 | 4.91 |
| −2.5 | 5.093 |
| 0 | 6.283 |
| 2.5 | 7.473 |
| 5 | 7.657 |
| 7.5 | 7.721 |
| 10 | 7.754 |
Sweep one variable
Variables and how each one contributes
Solution
What the equation gives you. Everything else feeds it.
Value of tan x
The value of tan x, a plain number. It can be any finite real value. Moves the principal angle, with most change near 0 and least for large values of u. Raising it by 0.25 (to −0.75) moves the result by +2.58 %, from 5.498 to 5.64 rad.
Whole number of half-turns
The whole number of half-turns added to the principal value, which can be negative. Adds π radians for every unit of k. Raising it by 1 (to 3) moves the result by +57.1 %, from 5.498 to 8.639 rad.
Worked example with the values above
- Principal value: arctan(−1) = −0.7854 rad.
- Add k π = 2 × 3.1416 = 6.2832.
- x = −0.7854 + 6.2832 = 5.4978 rad.
Assumptions and limits
- Angles in radians, and a principal value of the inverse function.
- For the sine rule, the sides and angle are consistent with a triangle that exists.
Common mistakes
- Giving only the calculator’s principal solution and missing the second one in the interval.
- Forgetting to add whole multiples of π for tangent or 2π for sine and cosine in a general solution.
- Using a period for the argument bx when x is measured in degrees.
Statistics and uncertainty
Each answer is exact but may sit on a boundary: near ±1 an inverse sine or cosine is very sensitive to rounding, and in the ambiguous case a small change in the data can add or remove a second triangle.
Where it is used
- Solving periodic problems: tides, alternating currents, vibration, rotating machinery.
- Solving triangles from two sides and a non-included angle.
Maths you need
- Inverse trigonometric functions: arcsin, arccos and arctan
- Period: the length of one full cycle
- Sine rule: a/sin A = b/sin B