Trigonometry — tables, formulas and equations explained step by step
Basic trigonometric equation
Pick a function and enter the right-hand side to get the general formula for the roots and an explanation of how it was obtained. The answer is written the way a teacher expects it: with n ∈ ℤ.
How it was solved — step by step
On the graph: why there are infinitely many roots
The orange line y = a crosses the wave at the solution points. Sine and cosine repeat every 2π (this is the period), so the intersections form an infinite chain — each period adds the same set of roots. That is where the “+ 2πn” or “+ πn” in the answer comes from.
On the unit circle
The angle is measured from the cos axis counterclockwise. The cosine of the angle is the horizontal coordinate of the point, the sine is the vertical one. The solutions of sin x = a are the points whose height equals a (a horizontal line), and cos x = a are the points with the required horizontal coordinate (a vertical line). Usually there are two such points on the circle — hence two roots per revolution.
Function values for an angle
Enter an angle in degrees (for example 60) or in radians (for example π/3 or pi/3) to get sin, cos, tan and cot. For table angles the answer is exact, otherwise it is rounded to 6 digits.
| sin | |
| cos | |
| tg | |
| ctg |
Inverse functions
arcsin, arccos, arctan and arccot return the angle from a function value — the principal value, in degrees and radians. arcsin and arccos are defined only for values from −1 to 1.
Table of values
Values for the main angles — the ones asked about most often. A dash means the function is undefined at that point.
| Degrees | Radians | sin | cos | tg | ctg |
|---|
Core formulas
Identities
| sin²x + cos²x = 1 | the fundamental identity |
| tg x = sin x / cos x | ctg x = cos x / sin x |
| tg x · ctg x = 1 | 1 + tg²x = 1/cos²x |
Sum of arguments
| sin(α ± β) = sin α·cos β ± cos α·sin β |
| cos(α ± β) = cos α·cos β ∓ sin α·sin β |
| tg(α ± β) = (tg α ± tg β) / (1 ∓ tg α·tg β) |
Double and half angle
| sin 2α = 2·sin α·cos α | |
| cos 2α = cos²α − sin²α = 1 − 2sin²α = 2cos²α − 1 | |
| sin²α = (1 − cos 2α)/2 | cos²α = (1 + cos 2α)/2 |
Reduction formulas
The rule: for π/2 and 3π/2 the function switches to its cofunction (sin ↔ cos, tan ↔ cot); for π and 2π it stays the same. The sign is taken from the quadrant the original angle lies in.
| sin(π/2 − α) = cos α | cos(π/2 − α) = sin α |
| sin(π − α) = sin α | cos(π − α) = −cos α |
| sin(π + α) = −sin α | cos(π + α) = −cos α |
| sin(2π − α) = −sin α | cos(2π − α) = cos α |
📚 Theory: how to solve basic equations
A basic trigonometric equation is an equality of the form sin x = a, cos x = a or tan x = a. Solving it means finding all angles at which the function equals a. There are infinitely many such angles: the functions are periodic, so the answer is written as a general formula with an integer parameter n.
The period of the sine is 2π, but on a period its roots are symmetric about π/2, which is why the formula contains the factor (−1)ⁿ and the period πn. The roots of the cosine are symmetric about zero — hence the ± sign. The tangent has period π and the simplest formula of all.
The special cases sin x = 0, ±1 and cos x = 0, ±1 are memorised separately. The general formula works for them too, but it is bulky, and the short form saves time during an exam.
If |a| > 1, an equation with a sine or a cosine has no solutions: these functions never leave the range from −1 to 1. The tangent and cotangent have no such restriction — they take any value.