# حاسبة المثلثات: sin وcos وtan ودوالها العكسية

> احسب sin وcos وtan وsec وcsc وcot بالدرجات أو الراديان أو الغراد، بدقة عند مضاعفات 15°، مع جميع حلول الدوال العكسية.

النسخة التفاعلية: https://www.calcopenly.com/ar/geometry/trig-functions-calculator
الموضوع: حاسبات الهندسة

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.

## المدخلات

- **I want** (الخيارات: Function of an angle, Angle from a value)
- **Function** (الخيارات: sin, cos, tan, sec, csc, cot)
- **الزاوية**: Expressions work, e.g. pi/4 with the unit set to radians
- **Inverse function** (الخيارات: arcsin, arccos, arctan, arcsec, arccsc, arccot)
- **القيمة**
- **Angle unit** (الخيارات: درجات, راديان, غراديان)

## النتائج

- النتيجة — النتيجة الرئيسية
- القيمة الدقيقة
- sin
- cos
- tan
- sec
- csc
- cot
- Angle in degrees (°)
- Angle in radians (rad)
- Reference angle (°)
- Position
- All solutions

## الصيغة

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

## أمثلة محلولة

### sin 30°

- I want: Function of an angle
- Function: sin
- الزاوية: 30
- Angle unit: درجات
- **النتيجة: 0.5**
- **القيمة الدقيقة: 1/2**
- **cos: 0.8660254038**
- مصدر التحقق: ⁨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
- الزاوية: 45
- Angle unit: درجات
- **النتيجة: 1**
- **القيمة الدقيقة: 1**
- **sec: 1.4142135624**
- مصدر التحقق: ⁨Standard special value; Python 3.8 math: 1/cos(pi/4)⁩

### cos 120° (second quadrant)

- I want: Function of an angle
- Function: cos
- الزاوية: 120
- Angle unit: درجات
- **النتيجة: -0.5**
- **القيمة الدقيقة: −1/2**
- **Reference angle: 60 °**
- **Position: Quadrant II**
- مصدر التحقق: ⁨cos(180° − 60°) = −cos 60° = −1/2⁩

### sin(π/4) in radians

- I want: Function of an angle
- Function: sin
- الزاوية: pi/4
- Angle unit: راديان
- **النتيجة: 0.7071067812**
- **القيمة الدقيقة: √2/2**
- **Angle in degrees: 45 °**
- مصدر التحقق: ⁨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
- الزاوية: 200
- Angle unit: غراديان
- **النتيجة: -1**
- **sin: 0**
- **tan: 0**
- **Position: On the negative x-axis**
- مصدر التحقق: ⁨200 grad = 180° exactly (400 grad per turn)⁩

### csc 15°

- I want: Function of an angle
- Function: csc
- الزاوية: 15
- Angle unit: درجات
- **النتيجة: 3.8637033052**
- **القيمة الدقيقة: √6+√2**
- مصدر التحقق: ⁨Python 3.8 math: 1/sin(radians(15)) and sqrt(6)+sqrt(2)⁩

## الأسئلة

### 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.

### ما مدى دقة «⁨حاسبة المثلثات: sin وcos وtan ودوالها العكسية⁩»؟

تعتمد الدقة على مدخلاتك وافتراضات الطريقة. يستخدم الحساب العشري 50 رقمًا معنويًا، لكن التقديرات والأساليب العددية وبيانات المصدر قد تكون أقل دقة؛ تقريب القيم المعروضة لا يزيل هذه الحدود. أمثلة محلولة جرى التحقق منها بمصادر مستقلة: 13. مثلًا، يجري التحقق من «⁨sin 30°⁩» بالرجوع إلى ⁨Standard special value (A&S Table 4.3); Python 3.8 math: cos(radians(30))⁩.

### ما مصدر هذه الطريقة؟

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.

## المصادر

- [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