# Тригонометрический калькулятор: sin, cos, tan и обратные

> sin, cos, tan, sec, csc и cot в градусах, радианах или градах, точные значения для кратных 15° и обратные функции со всеми решениями.

Интерактивная версия: https://www.calcopenly.com/ru/geometry/trig-functions-calculator
Тема: Геометрические калькуляторы

Enter an angle in degrees, radians or gradians to get all six trigonometric functions. Sine and cosine are the coordinates of the matching point on the unit circle; the rest follow as tan = sin/cos, sec = 1/cos, csc = 1/sin and cot = cos/sin. At multiples of 15° the exact surd form is shown as well, such as sin 60° = √3/2. The inverse mode turns a value back into an angle and lists every angle that shares it.

The default, sin 30°, is exactly 1/2, and the drawing puts the point at (0.866, 0.5). The same functions convert between angles and slopes: a roof pitched at 37° rises tan 37° = 0.7536 m per metre of run.

Principal values follow the NIST Digital Library of Mathematical Functions: arcsin and arctan return −90° to 90°, arccos 0° to 180°. arccot uses the continuous range 0° to 180°, which differs from the DLMF definition for negative inputs.

## Входные данные

- **I want** (варианты: Function of an angle, Angle from a value)
- **Function** (варианты: sin, cos, tan, sec, csc, cot)
- **Угол**: Expressions work, e.g. pi/4 with the unit set to radians
- **Inverse function** (варианты: arcsin, arccos, arctan, arcsec, arccsc, arccot)
- **Значение**
- **Angle unit** (варианты: Градусы, Радианы, Грады)

## Результаты

- Результат — основной результат
- Точное значение
- sin
- cos
- tan
- sec
- csc
- cot
- Angle in degrees (°)
- Angle in radians (rad)
- Reference angle (°)
- Position
- All solutions

## Формула

$$
\sec\theta = \frac{1}{\cos\theta},\quad \csc\theta = \frac{1}{\sin\theta},\quad \cot\theta = \frac{\cos\theta}{\sin\theta},\quad 1^\circ = \frac{\pi}{180}\,\text{rad} = \frac{10}{9}\,\text{grad}
$$

## Примеры с решением

### sin 30°

- I want: Function of an angle
- Function: sin
- Угол: 30
- Angle unit: Градусы
- **Результат: 0.5**
- **Точное значение: 1/2**
- **cos: 0.8660254038**
- Источник проверки: Standard special value (A&S Table 4.3); Python 3.8 math: cos(radians(30))

### tan 45°

- I want: Function of an angle
- Function: tan
- Угол: 45
- Angle unit: Градусы
- **Результат: 1**
- **Точное значение: 1**
- **sec: 1.4142135624**
- Источник проверки: Standard special value; Python 3.8 math: 1/cos(pi/4)

### cos 120° (second quadrant)

- I want: Function of an angle
- Function: cos
- Угол: 120
- Angle unit: Градусы
- **Результат: -0.5**
- **Точное значение: −1/2**
- **Reference angle: 60 °**
- **Position: Quadrant II**
- Источник проверки: cos(180° − 60°) = −cos 60° = −1/2

### sin(π/4) in radians

- I want: Function of an angle
- Function: sin
- Угол: pi/4
- Angle unit: Радианы
- **Результат: 0.7071067812**
- **Точное значение: √2/2**
- **Angle in degrees: 45 °**
- Источник проверки: Python 3.8 math: sin(pi/4) = √2/2

### cos of 200 grad (negative x-axis, edge case)

- I want: Function of an angle
- Function: cos
- Угол: 200
- Angle unit: Грады
- **Результат: -1**
- **sin: 0**
- **tan: 0**
- **Position: On the negative x-axis**
- Источник проверки: 200 grad = 180° exactly (400 grad per turn)

### csc 15°

- I want: Function of an angle
- Function: csc
- Угол: 15
- Angle unit: Градусы
- **Результат: 3.8637033052**
- **Точное значение: √6+√2**
- Источник проверки: Python 3.8 math: 1/sin(radians(15)) and sqrt(6)+sqrt(2)

## Вопросы

### How do you convert degrees to radians?

Multiply by π/180: 30° is π/6 ≈ 0.5236 rad, 45° is π/4 ≈ 0.7854 rad and 180° is π rad. To go back, multiply radians by 180/π ≈ 57.2958. Mixing the two is the most common trig mistake: sin 30 with the angle read as radians is −0.988, not 0.5.

### What are the exact values of sin, cos and tan at 30°, 45° and 60°?

sin 30° = 1/2, sin 45° = √2/2 ≈ 0.7071 and sin 60° = √3/2 ≈ 0.8660; cosine takes the same values in reverse order, so cos 30° = √3/2 and cos 60° = 1/2. tan 30° = √3/3 ≈ 0.5774, tan 45° = 1 and tan 60° = √3 ≈ 1.7321. All of them come from the 30-60-90 and 45-45-90 triangles.

### Why is tan 90° undefined?

Because tan θ = sin θ / cos θ and cos 90° = 0, so the ratio divides by zero. As θ approaches 90° from below, tan θ grows without bound: tan 89° ≈ 57.29 and tan 89.9° ≈ 572.96. The same happens at 270° and every 90° + 180°k. Secant is undefined at those angles too, and cosecant and cotangent wherever sin θ = 0.

### Is sin⁻¹ the same as 1/sin?

No. sin⁻¹ x, also written arcsin x, is the inverse function: the angle whose sine is x, so sin⁻¹ 0.5 = 30°. 1/sin x is the reciprocal, called cosecant: csc 30° = 1/0.5 = 2. The clash arises because sin² x means (sin x)², while the −1 exponent on a function name means its inverse; writing arcsin avoids the ambiguity.

### Why does arcsin give only one angle?

A function must return a single value, so arcsin is restricted to a principal range of −90° to 90° (DLMF §4.23). Every other angle with the same sine follows from θ and 180° − θ plus whole turns: sin θ = 0.5 for θ = 30° + 360°k or 150° + 360°k, with k any integer. For arccos the range is 0° to 180° and the other family is −θ + 360°k.

### Насколько точен «Тригонометрический калькулятор: sin, cos, tan и обратные»?

Точность зависит от введённых данных и допущений метода. Десятичная арифметика использует 50 значащих цифр, но оценки, численные методы и исходные данные могут быть менее точными; округление на экране не устраняет эти ограничения. Решённые примеры, проверенные по независимым источникам: 13. Например, «sin 30°» проверяется по источнику Standard special value (A&S Table 4.3); Python 3.8 math: cos(radians(30)).

### Откуда взята методика?

NIST Digital Library of Mathematical Functions §4.14 (definitions) and §4.23 (inverse trigonometric functions, principal values); Abramowitz & Stegun, Handbook of Mathematical Functions, Table 4.3 — special values of the trigonometric functions.

## Источники

- [NIST Digital Library of Mathematical Functions §4.14 (definitions) and §4.23 (inverse trigonometric functions, principal values)](https://dlmf.nist.gov/4.23)
- Abramowitz & Stegun, Handbook of Mathematical Functions, Table 4.3 — special values of the trigonometric functions