5-6 Inverse Trig Functions: DifferentiationObjective: Develop properties of the 6 inverse trig functions and differentiate an inverse trig function. Ms. Battaglia AP Calculus
Graphs 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
Evaluating Inverse Trig Functions a. b. c. d.
Properties of 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.
Solving an Equation arctan(2x – 3) = π/4
Using Right Triangles • Given y = arcsinx, where 0 < y < π/2, find cos y. • Given y = arcsec( ), find tan y.
Derivatives of Inverse Trig Functions Let u be a differentiable function of x.
Differentiating Inverse Trig Functions a. b. c. d.
Maximizing an Angle 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?
See Page 378 for a Review of Basic Differentiation Rules for Elementary Functions.
Classwork/Homework • AB: Read 5.6 Page 379 #5-11 odd, 17, 27, 29, 43-51 odd • BC: AP Sample