Aim: Differentiate Inverse Trig Functions Course: Calculus Do Now: Aim: How do we differentiate Inverse Trig functions? Does y = sin x have an inverse?

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Presentation transcript:

Aim: Differentiate Inverse Trig Functions Course: Calculus Do Now: Aim: How do we differentiate Inverse Trig functions? Does y = sin x have an inverse? yes, but only in restricted domain.

Aim: Differentiate Inverse Trig Functions Course: Calculus Definition of Inverse Trig Functions arcsinx  sin -1 x alternate notation ‘the angle whose sin is...’

Aim: Differentiate Inverse Trig Functions Course: Calculus Graphs of Inverse Trig Functions

Aim: Differentiate Inverse Trig Functions Course: Calculus Graphs of Inverse Trig Functions y = arccsc x y = arcsec x y = arctan x

Aim: Differentiate Inverse Trig Functions Course: Calculus Evaluating Inverse Trig Functions ‘the angle whose sin is -1/2’ sin is negative in QIII and IV however, restricted domain reference angle of  /6 angle y is -  /6  0º30º45º60º90º sin  0 cos  1 tan  UND.

Aim: Differentiate Inverse Trig Functions Course: Calculus Evaluating Inverse Trig Functions ‘the angle whose cos is 0’ reference angle of  /2 angle y is  /2 however, restricted domain  0º30º45º60º90º sin  0 cos  1 tan  UND.

Aim: Differentiate Inverse Trig Functions Course: Calculus Evaluating Inverse Trig Functions reference angle of  /3 angle y is  /3 ‘the angle whose tan is ’ however, restricted domain  0º30º45º60º90º sin  0 cos  1 tan  UND.

Aim: Differentiate Inverse Trig Functions Course: Calculus Using Inverses to Solve an Equation take tan of both sides inverse property x = 2solve for x

Aim: Differentiate Inverse Trig Functions Course: Calculus Using Right Triangles Given y = arcsin x, where 0 < y <  /2, find cos y. Since y = arcsin x, then sin y = x opposite x y 1

Aim: Differentiate Inverse Trig Functions Course: Calculus Using Right Triangles opposite y adjacent Since y = arcsec x, then sec y = x 1 2

Aim: Differentiate Inverse Trig Functions Course: Calculus Derivatives: Inverse Trig Functions derivative of inverse sine

Aim: Differentiate Inverse Trig Functions Course: Calculus Derivatives of Inverse Trig Functions

Aim: Differentiate Inverse Trig Functions Course: Calculus Model Problems u = 2x u = 3x u = x 1/2 u = e 2x

Aim: Differentiate Inverse Trig Functions Course: Calculus Model Problems u = x 2 u = 5x u = x 2 - x

Aim: Differentiate Inverse Trig Functions Course: Calculus Elementary Functions Elementary functions have proven sufficient to model most phenomena in physics, chemistry, biology, engineering, economics, etc. An elementary function is a function from the following list or one that can be formed as a sum, product, quotient or composition of functions in the list. Algebraic Polynomial Rational Functions w/radicals Transcendental Logarithmic Exponential Trigonometric Inverse Trigonometric

Aim: Differentiate Inverse Trig Functions Course: Calculus Using Right Triangles  0º30º45º60º90º sin  0 cos  1 tan  UND.