maths.free › Trigonometry › Inverse trigonometric functions

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.

कार्यरत उदाहरण · derivative of arctan(x)

Calculus · Derivatives

Differentiate atan(x)

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

चरणानुक्रमे

  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).

उत्तर उघडा
f'(x) = \frac{1}{x^{2} + 1}

आता तुम्ही प्रश्न निवडा, किंवा स्वतःचा प्रश्न टाइप करा किंवा रेखाटून दाखवा. प्रत्येक टप्प्यात, चित्रे आहेत, तुम्ही विचारण्याआधी उत्तर लपवलेले आहे.

स्वतःचे कार्य चालू ठेवा

मोफत खाते प्रत्येक शिकवणी वर टिपण्णी जोडते, तुम्ही पूर्ण केलेल्या गोष्टीचा नोंदवही, तुमच्या सोडविलेल्या समस्या एका ठिकाणी, आणि या पानाच्या विषयी तुम्ही विचारू शकता अशा शिक्षकाला. गणित स्वतःच सर्वांसाठी खुले आहे, दाखलन झालेले असो किंवा नसो.

नोंदणी करा दाखलन

येथे वापरलेले चिन्ह

Tap any symbol for the full definition, a picture, and what every letter in it means.

कसे कराल: Inverse trigonometric functions

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

लोक विचारतात ते प्रश्न

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.

अधिक माहिती 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