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Inverse trigonometric functions

arcsin, arccos, arctan (which angle has this ratio) and their derivatives.

sin is not one-to-one, so its inverse is defined on a restricted range: arcsin returns an angle in [−π/2, π/2], arccos in [0, π], arctan in (−π/2, π/2). Picture it: reflect the sine curve across y = x and keep one branch. Think it: their derivatives are algebraic (1/√(1 − x²), 1/(1 + x²)), which is why they appear as antiderivatives of ordinary fractions.

Çözümlü örnek · derivative of arctan(x)

Calculus · Derivatives

Differentiate atan(x)

\frac{d}{dx}\left[\operatorname{atan}{\left(x \right)}\right]

Adım adım

  1. \frac{d}{dx}\left[\operatorname{atan}{\left(x \right)}\right]

    Start from the derivative to compute.

  2. \frac{1}{x^{2} + 1}

    (arctan u)′ = 1/(1+u²).

Cevabı göster
f'(x) = \frac{1}{x^{2} + 1}

Şimdi sen Bir problem seçin, ya da kendi probleminizi yazın ya da çizin. Her adım, bir resim, cevap sorulana kadar gizli.

Kendi çalışmalarınızı saklayın

Ücretsiz bir hesap her ders için notlar, bitirdiğiniz şeylerin kaydı, çözüldüğünüz problemlerin bir yerdeki kaydı ve bu sayfa hakkında soru sormak için bir öğretmen ekler. Matematik kendisi, giriş yapmış veya yapmamış herkese açıktır.

Kayıt Ol Giriş

Burada kullanılan sembolleri

Tüm tanım, resim ve her harfin anlamı için herhangi bir sembolü tıklayın.

Nasıl: Inverse trigonometric functions

  1. Start from the derivative to compute.
  2. (arctan u)′ = 1/(1+u²).

Sık sorulan sorular

Why radians instead of degrees?

A radian is the angle whose arc equals the radius, so it is a pure ratio rather than an arbitrary 1/360 of a turn. With radians the derivative of sin x is exactly cos x; with degrees a factor of π/180 appears everywhere.

Why does sin x = 1/2 have infinitely many solutions?

Sine repeats every full turn, and within one turn it reaches 1/2 twice (at π/6 and 5π/6). Add any whole number of turns to either and it is still true.

How do I remember the exact values?

Two triangles: the 45-45-90 with sides 1, 1, √2 and the 30-60-90 with sides 1, √3, 2. Every value for 30°, 45° and 60° is a ratio of those sides; symmetry gives the rest of the circle.

More in Trigonometry

The unit circleTrigonometric equationsTrigonometric identitiesDegrees and radiansRight-triangle trigonometry (SOH-CAH-TOA)Law of sines and law of cosinesGraphs of sine, cosine and tangent