TrigPrecalc Chapter 4 7 Inverse trig functions n

Trig/Precalc Chapter 4. 7 Inverse trig functions n Objectives n Evaluate and graph the inverse sine function n Evaluate and graph the remaining five inverse trig functions n 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 n The sine function gives us ratios representing opposite over hypotenuse in all 4 quadrants. π/2 1 n The inverse sine gives us the angle or arc length on the unit circle that has the given ratio. -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 ______ No Solution c. sin-1(2) = _____ Sin domain is [-1, 1], therefore No solution 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 n y = arcsin x n Domain: [-1, 1] n Range: n y = arccos x n Domain: [ -1, 1] n Range: n y = arctan x n Domain: (-∞, ∞) n Range: y = Arcsin (x) y = Arccos (x) 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

Guided practice Example of [E] 28 & 30 Use an inverse trig function “θ as a function of x” means to write an equation to write θ as a function of x. of the form θ equal to an expression with x in it. 28. Cos θ = 4/x so θ = cos-1(4/x) where x > 0 30. tan θ = (x – 1)/(x 2 – 1) θ = tan-1(x – 1)/(x 2 – 1) where x – 1 > 0 , x > 1 12

Composition of trig functions Find the exact value, sketch a triangle. cos(tan-1 (2)) = _____ This means tan θ = 2 so… draw the triangle Label the adjacent and opposite sides √ 5 2 θ 1 Find the hypo. using Pyth. Theorem So the 13

Example Write an algebraic expression that is equivalent to the given expression. cot(arctan(1/x)) 1) Draw and label the triangle 1 ---(let u be the unknown angle) 2) Use the Pyth. Theo. to compute the hypo 3) Find the cot of u u x 14

You Try! Evaluate: -4/3 0 rad. csc[arccos(-2/3)] (Hint: Draw a triangle) Rewrite as an algebraic expression:

A L E K S Word problem involving sin or cos function: P type 1 An object moves in simple harmonic motion with amplitude 12 cm and period 0. 1 seconds. At time t = 0 seconds , its displacement d from rest is 12 in a negative direction, and initially it moves in a negative direction. Give the equation modeling the displacement d as a function of time t. Clear Next >> Undo Explain pcalc 643 Help

A L E K S Word problem involving sin or cos function: P type 2 The depth of the water in a bay varies throughout the day with the tides. Suppose that we can model the depth of the water with the following function. h(t) = 13 + 6. 5 sin 0. 25 t In this equation, h(t) is the depth of the water in feet, and t is the time in hours. Find the following. If necessary, round to the nearest hundredth. Frequency of h: cycles per hour Period of h: hours Minimum depth of the water: feet Next >> Clear Undo Explain pcalc 643 Help