TrigPrecalc Chapter 5 7 Inverse trig functions Objectives

Trig/Precalc Chapter 5. 7 Inverse trig functions • Objectives • Evaluate and graph the inverse sine function • Evaluate and graph the remaining five inverse trig functions • Evaluate and graph the composition of trig functions 1

The basic sine function fails the horizontal line test. It is not one-to-one so we can’t find an inverse function unless we restrict the domain. Highlight the curve –π/2 < x < π/2 y = sin(x) -π/2 On the interval [-π/2, π/2] for sin x: the domain is [-π/2, π/2] and the range is [-1, 1] Therefore π/2 π 2π We switch x and y to get inverse functions So for f(x) = sin-1 x the domain is [-1, 1] and range is [-π/2, π/2] 2

Graphing the Inverse First we draw the sin curve When we get rid of all the duplicate numbers we get this curve Next we rotate it across the y=x line producing this curve This gives us: Domain : [-1 , 1] Range: 3

Inverse sine function y = sin-1 x or y = arcsin x • The sine function gives us ratios representing opposite over hypotenuse in all 4 quadrants. • The inverse sine gives us the angle or arc length on the unit circle that has the given ratio. π/2 1 -π/2 Remember the phrase “arcsine of x is the angle or arc whose sine is x”. 4

Evaluating Inverse Sine If possible, find the exact value. a. arcsin(-1/2) = ____ We need to find the angle in the range [-π/2, π/2] such that sin y = -1/2 What angle has a sin of ½? _______ What quadrant would it be negative and within the range of arcsin? ____ Therefore the angle would be ______ 5

Evaluating Inverse Sine cont. b. sin-1( ) = ____ We need to find the angle in the range [-π/2, π/2] such that sin y= √ 3 2 What angle has a sin of ? _______ 1 What quadrant would it be positive and within the range of arcsin? ____ Therefore the angle would be ______ c. sin-1(2) = _____ 6

Graphs of Inverse Trigonometric Functions The basic idea of the arc function is the same whether it is arcsin, arccos, or arctan 7

Inverse Functions Domains and Ranges • y = arcsin x y = Arcsin (x) • Domain: [-1, 1] • Range: • y = arccos x • Domain: [ -1, 1] • Range: y = Arccos (x) • y = arctan x • Domain: (-∞, ∞) • Range: y = Arctan (x) 8

Evaluating Inverse Cosine If possible, find the exact value. a. arccos(√(2)/2) = ____ We need to find the angle in the range [0, π] such that cos y = √(2)/2 What angle has a cos of √(2)/2 ? _______ What quadrant would it be positive and within the range of arccos? ____ Therefore the angle would be ______ b. cos-1(-1) = __ What angle has a cos of -1 ? _______ 9

Warnings and Cautions! Inverse trig functions are equal to the arc trig function. Ex: sin-1 θ = arcsin θ Inverse trig functions are NOT equal to the reciprocal of the trig function. Ex: sin-1 θ ≠ 1/sin θ There are NO calculator keys for: sec-1 x, csc-1 x, or cot 1 x And csc-1 x ≠ 1/csc x sec-1 x ≠ 1/sec x cot-1 x ≠ 1/cot x 10

Evaluating Inverse functions with calculators ([E] 25 & 34) If possible, approximate to 2 decimal places. 19. arccos(0. 28) = ____ 22. arctan(15) = _____ 26. cos-1(0. 26) = ____ 34. tan-1(-95/7) = ____ Use radian mode unless degrees are asked for. 11

You Try! Evaluate: csc[arccos(-2/3)] (Hint: Draw a triangle)