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Trig/Precalc Chapter 4.7 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. y = sin(x). - π /2. π /2. π. 2 π. - PowerPoint PPT Presentation

*Trig/PrecalcChapter 4.7 Inverse trig functionsObjectivesEvaluate and graph the inverse sine functionEvaluate and graph the remaining five inverse trig functionsEvaluate and graph the composition of trig functions

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

*Graphing the InverseWhen we get rid of all the duplicate numbers we get this curveNext we rotate it across the y=x line producing this curveFirst we draw the sin curve

*Inverse sine function y = sin-1 x or y = arcsin xThe 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. Remember the phrase arcsine of x is the angle or arc whose sine is x./2-/21

*Evaluating Inverse Sine

If possible, find the exact value.arcsin(-1/2) = ____

We need to find the angle in the range [-/2, /2] such that sin y = -1/2

What angle has a sin of ? _______What quadrant would it be negative and within the range of arcsin? ____Therefore the angle would be ______

*Evaluating Inverse Sine cont.b. sin-1( ) = ____ We need to find the angle in the range [-/2, /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 solutionNo Solution

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

*Inverse Functions Domains and Rangesy = arcsin xDomain: [-1, 1]Range:

y = arccos xDomain: [ -1, 1]Range:

y = arctan xDomain: (-, )Range:y = Arccos (x)

*Evaluating Inverse Cosine

If possible, find the exact value.arccos((2)/2) = ____ We need to find the angle in the range [0, ] 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-1 = arcsin

Inverse trig functions are NOT equal to the reciprocal of the trig function. Ex: sin-1 1/sin

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 & 30Use an inverse trig functionto write as a function of x. 28. Cos = 4/x so = cos-1(4/x) where x > 0

30. tan = (x 1)/(x2 1) = tan-1(x 1)/(x2 1) where x 1 > 0 , x > 1 as a function of x means to write an equation of the form 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 = 2 sodraw the triangle Label the adjacent and opposite sides

Find the hypo. using Pyth. Theorem

So the215

*ExampleWrite an algebraic expression that is equivalent to the given expression. cos(arctan(1/x))ux 11) 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

You Try! Evaluate: -4/3

0 rad.

csc[arccos(-2/3)] (Hint: Draw a triangle)

Rewrite as an algebraic expression:

Word problem involving sin or cos function: P type 1pcalc643ALEKSAn 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.UndoHelpClearNext >>Explain

Word problem involving sin or cos function: P type 2pcalc643ALEKSThe 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 0.25tIn 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: feetUndoHelpClearNext >>Explain