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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.

Үйлдэл · derivative of arctan(x)

Calculus · Derivatives

Differentiate atan(x)

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

Алхам алхмаар

  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).

Хариулт
f'(x) = \frac{1}{x^{2} + 1}

Одоо та Даалгавар сонго, эсвэл бич, эсвэл зур. Бүх алхам, зураг, хариулт нь асуулт асуух хүртэл нуугдаж байдаг.

Өөрийнхөө ажлыг үргэлжлүүл

Үнэгүй бүртгэл нь хичээл бүрийн тэмдэглэл, төгссөн хичээл, шийдсэн даалгаврын тэмдэглэл, энэ хуудсан дээр асууж болох багшийг нэгтгэж өгдөг. Математик нь бүртгэлтэй, бүртгэлтэйгүй хэн бүхэнд нээлттэй.

Бүртгүүлэх Нэвтрэх

Энд ашигласан тэмдэгтүүд

Бүх тодорхойлолт, зураг, үсгийн утгыг харахын тулд ямар ч тэмдэгтийг дарна уу.

Яаж: Inverse trigonometric functions

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

Хүмүүс асуудаг асуултууд

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.

Бүх зүйл 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