5-6 Inverse Trig Functions: Differentiation

Objective: Develop properties of the 6 inverse trig functions and differentiate an inverse trig function.

Ms. BattagliaAP Calculus

Function Domain RangeDefinitions of Inverse Trig Functions

y = arcsinx y = arccosx

Graphs of Inverse Trig Functions

y = arctanx y = arccscx

Graphs of Inverse Trig Functions

y = arcsecx y = arccotx

Graphs of Inverse Trig Functions

a. b.

c. d.

Evaluating Inverse Trig Functions

If -1 < x < 1 and –π/2 < y < π/2 then

sin(arcsinx) = x and arcsin(siny) = y

If –π/2 < y < π/2, then

tan(arctanx) = x and arctan(tany) = y

If |x| > 1 and 0 < y < π/2 or π/2 < y < π, then

Sec(arcsecx) = x and arcsec(secy) = y.

Similar properties hold for other inverse trig functions.

Properties of Inverse Trig Functions

arctan(2x – 3) = π/4Solving an Equation

a. Given y = arcsinx, where 0 < y < π/2, find cos y.b. Given y = arcsec( ), find tan y.

Using Right Triangles

Let u be a differentiable function of x.

Derivatives of Inverse Trig Functions

a.

b.

c.

d.

Differentiating Inverse Trig Functions

A Derivative That Can Be Simplified

A photographer is taking a picture of a painting hung in an art gallery. The height of the painting is 4 ft. The camera lens is 1 ft below the lower edge of the painting. How far should the camera be from the painting to maximize the angle subtended by the camera lens?

Maximizing an Angle

See Page 378 for a Review of Basic Differentiation Rules for Elementary Functions.

AB: Read 5.6 Page 379 #5-11 odd, 17, 27, 29, 43-51 odd

BC: AP Sample

Classwork/Homework