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Calculadora trigonométrica: sin, cos, tan e inversas

sin, cos, tan, sec, csc y cot en grados, radianes o grados centesimales; valores exactos en múltiplos de 15° e inversas con todas sus soluciones.

Actualizado Ejemplos verificados: 13

Expressions work, e.g. pi/4 with the unit set to radians
Probar
Resultado
Resultado: 0.5
Máximo de decimales: 10; Al más cercano; empates alejándose de cero
Valor exacto
1/2
cos
0.8660254038
tan
0.5773502692
sec
1.1547005384
csc
2
cot
1.7320508076
Angle in radians
0.5235987756rad
Reference angle
30°
Position
Quadrant I

sin 30° = 0.5 (exactly 1/2). The angle sits in Quadrant I, where the point on the unit circle is (0.866, 0.5).

Unit circle

sin = 0.5cos = 0.86630°(0.866, 0.5)
All six functions at 30° Filas: 6
FunctionValorExacto
sin0.51/2
cos0.866√3/2
tan0.5774√3/3
sec1.15472√3/3
csc22
cot1.7321√3
Cómo se calcula S

  1. Ángulo

    30∘30^\circ

  2. Position on the unit circle

    Quadrant I; reference angle 30°.

  3. Sine and cosine

    sin⁡θ=0.5,cos⁡θ=0.8660254038\sin\theta = 0.5,\qquad \cos\theta = 0.8660254038

  4. sin from sine and cosine

    sin⁡θ=0.5\sin\theta = 0.5

    Exact: sin 30° = 1/2.

Acerca de Calculadora trigonométrica: sin, cos, tan e inversas

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.

Ejemplos resueltos

sin 30°

I want
Function of an angle
Function
sin
Ángulo
30
Angle unit
Grados
Resultado
0.5
Valor exacto
1/2
cos
0.8660254038

Fuente de comprobación: 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
Ángulo
45
Angle unit
Grados
Resultado
1
Valor exacto
1
sec
1.4142135624

Fuente de comprobación: Standard special value; Python 3.8 math: 1/cos(pi/4)

cos 120° (second quadrant)

I want
Function of an angle
Function
cos
Ángulo
120
Angle unit
Grados
Resultado
-0.5
Valor exacto
−1/2
Reference angle
60 °
Position
Quadrant II

Fuente de comprobación: cos(180° − 60°) = −cos 60° = −1/2

sin(π/4) in radians

I want
Function of an angle
Function
sin
Ángulo
pi/4
Angle unit
Radianes
Resultado
0.7071067812
Valor exacto
√2/2
Angle in degrees
45 °

Fuente de comprobación: Python 3.8 math: sin(pi/4) = √2/2

Preguntas

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.

¿Qué precisión tiene «Calculadora trigonométrica: sin, cos, tan e inversas»?

La precisión depende de tus datos y de los supuestos del método. El cálculo decimal usa 50 cifras significativas, pero las estimaciones, los métodos numéricos y los datos de origen pueden ser menos precisos; el redondeo mostrado no elimina esos límites. Ejemplos resueltos comprobados con fuentes independientes: 13. Por ejemplo, «sin 30°» se comprueba con Standard special value (A&S Table 4.3); Python 3.8 math: cos(radians(30)).

¿De dónde procede el método?

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.

Acerca de esta calculadora

sec⁡θ=1cos⁡θ,csc⁡θ=1sin⁡θ,cot⁡θ=cos⁡θsin⁡θ,1∘=π180 rad=109 grad\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}

Fuentes

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

Verificado con las referencias

Esta calculadora incluye 13 ejemplos resueltos con respuestas de fuentes independientes. Forman parte del conjunto de pruebas y también puedes ejecutarlos aquí.

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