Trigonometry — values, simplification and equations explained step by step
Problem
Type an expression, an equation or something to simplify, the way you would write it in your notebook. A number without a degree sign is read as degrees; use π for radians.
How it was solved — step by step
The other functions of this angle
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 “+ 2πk” or “+ πk” 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.
The angle is measured from the cos axis counterclockwise. The cosine is the horizontal coordinate of the point, the sine is its vertical coordinate.
Right triangle
Enter any two values, at least one of them a side, and the rest will be calculated.
sin α = a/c is the opposite leg over the hypotenuse, cos α = b/c is the adjacent leg over the hypotenuse, tan α = a/b is the opposite leg over the adjacent one.
How it was solved — step by step
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 k.
The sine has period 2π, and on each turn it takes the value a twice: at x = arcsin a and at x = π − arcsin a. That is why the answer has two series, each repeating every 2π. 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.
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