User:Hurricane Angel/Math 101

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Contents

[edit] Trigonometric Functions

[edit] Identities

[edit] Standard

[edit] Derivatives

{d \over dx}\ sin x = \cos x


{d \over dx}\ cos x = -\sin x


\tan x = \frac {sin x}{cos x}


{d \over dx}\ tan x = {d \over dx}\frac {sin x}{cos x} = \frac {(cos x)(cos x) - (-sin x)(sin x)}{cos^2 x} = \frac {cos^2 x + sin^2 x}{cos^2 x} *

  • Using the identity cos2x + sin2x = 1, it is revealed that;

{d \over dx}\ tan x = \frac {1}{cos^2 x} = \sec^2 x *

  • Secant function separately determined (below).


\cot x = \frac {cos x}{sin x}


{d \over dx}\ cot x ={d \over dx} \frac {cos x}{sin x} = \frac {(-sin x)(sin x) - (cos x)(cos x)}{sin^2 x} = \frac {-sin^2 x - cos^2 x}{sin^2 x}


\frac {(-1)(-sin^2 x - cos^2 x)}{(-1)(sin^2 x)} = \frac {1}{-sin^2 x} = -csc^2 x



{d \over dx}\ sec x = {d \over dx} \frac {1}{cos x} = \frac {(0)(cos x) - (-sin x)(1)}{cos^2 x} = \frac {sin x}{cos^2 x}


\frac {sin x}{cos^2 x} = \frac {sin x}{cos x} \cdot \frac {1}{cos x} = tan x sec x



{d \over dx}\ csc x = {d \over dx} \frac {1}{sin x} = \frac {(0)(sin x) - (cos x)(1)}{sin^2 x} = \frac {-cos x}{sin^2 x}


\frac {-cos x}{sin^2 x} = \frac {-cos x}{sin x} \cdot \frac {1}{sin x} = -cot x csc x


[edit] Integrals

[edit] Inverse Functions

[edit] Derivatives

{d \over dx}\ sin^{-1} x = \frac {1}{\sqrt {1-x^2}}

{d \over dx}\ cos^{-1} x = \frac {-1}{\sqrt {1-x^2}}

{d \over dx}\ tan^{-1} x = \frac {1}{1 + x^2}

{d \over dx}\ cot^{-1} x = \frac {-1}{1 + x^2}

{d \over dx}\ sec^{-1} x = \frac {1}{|x| \sqrt {x^2 - 1}}

{d \over dx}\ csc^{-1} x = \frac {-1}{|x| \sqrt {x^2 -1}}

[edit] Integrals

\int sin^{-1} x = x sin^{-1} x + \sqrt {1 - x^2} + C

\int tan^{-1} x = x tan^{-1} x - \frac {1}{2} ln(1 + x^2) + C

\int sec^{-1} x = x sec^{-1} x - ln |x + \sqrt {x^2 +1}| + C

[edit] Hyperbolic Functions

[edit] Derivatives

{d \over dx}\ sinh x = cosh x

{d \over dx}\ cosh x = sinh x

{d \over dx}\ tanh x = sec^2 x

{d \over dx}\ coth x = -csch^2 x

{d \over dx}\ sech x = -tanh sech x

{d \over dx}\ csch x = -coth csch x

[edit] Integrals