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*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