Trig/Precalc Chapter 4.7 Inverse trig functions

Trig/PrecalcChapter 4.7 Inverse trigfunctions • 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

y = sin(x) -π/2 π/2 π 2π 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 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] On the interval [-π/2, π/2] for sin x: the domain is [-π/2, π/2] and the range is [-1, 1] Therefore

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

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”.

Evaluating Inverse Sine If possible, find the exact value. • 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 ______

√3 2 1 Evaluating Inverse Sine cont. b. sin-1( ) = ____ We need to find the angle in the range [-π/2, π/2] such that sin y = What angle has a sin of ? _______ What quadrant would it be positive and within the range of arcsin? ____ Therefore the angle would be ______ c. sin-1(2) = _________ Sin domain is [-1, 1], therefore No solution No Solution

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

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

Evaluating Inverse Cosine If possible, find the exact value. • 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 ? _______

Warnings and Cautions! Inverse trig functions are equal to the arc trig function. Ex: sin-1θ = arcsin θ Inverse trig functions are NOT equal to the reciprocal of the trig function. Ex: sin-1θ ≠ 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

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.

Guided practice Example of [E] 28 & 30 Use an inverse trig function to write θ as a function of x. 28. Cos θ = 4/x so θ = cos-1(4/x) where x > 0 30. tan θ = (x – 1)/(x2 – 1) θ = tan-1(x – 1)/(x2 – 1) where x – 1 > 0 , x > 1 “θas a function of x” means to write an equation of the form θ equal to an expression with x in it.

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 Find the hypo. using Pyth. Theorem So the √5 2 θ 1

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

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

ALEKS 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 Undo Help Next >> Explain pcalc643

ALEKS 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 0.25t 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 Clear Undo Help Next >> Explain pcalc643

Trig/Precalc Chapter 4.7 Inverse trig functions