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4.8 Inverse Trigonometric Functions

The inverse trig functions. Inverse trig functions are not ?true" inverses? Instead, each inverse trig function has a domain that is restricted to intervals where the function is one-to-one and where they have the same range as the ordinary trig functions.Remember from Section 2.5, functions that

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4.8 Inverse Trigonometric Functions

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    1. 4.8 Inverse Trigonometric Functions Chris Reed

    2. The inverse trig functions Inverse trig functions are not true inverses Instead, each inverse trig function has a domain that is restricted to intervals where the function is one-to-one and where they have the same range as the ordinary trig functions. Remember from Section 2.5, functions that have an inverse are precisely one-to-one!

    3. Review: Properties of Inverse Functions Suppose that f is a one-to-one function: The inverse function f -1 is unique The domain of f -1 is the range of f The range of f -1 is the domain of f If x is in the domain of f -1 and y is in the domain of f, then: f -1 (x) = y if and only if f(y) = x

    4. Properties of Inverse Functions Continued If x is in the domain of f -1, then f(f -1(x))=x If x is in the domain of f, then f -1(f(x))=x The graph of y = f -1(x) is the reflection of the graph of y=f(x) about the line y=x.

    5. Personal Observation Notice that f -1 is NOT the same thing as 1/f for trig functions! We will start by examining the inverse of the sine function First though, is the sine function one-to-one? No, so we must restrict the domain!

    6. Suppose we look at the interval [-p/2, p/2]. Is this interval one-to-one? Yes, and notice that it also assumes all the values in its range, [-1, 1]. This restricted function has an inverse, which we will call the inverse sine, or arcsine function.

    7. Arcsine Function The arcsine function is denoted either as arcsin or as sin-1. We will (usually) use the arcsin notation to avoid confusion between the inverse sine function and the reciprocal of the sine function, csc x = (sin x)-1.

    8. Arcsine Function The arcsine function, denoted arcsin, has domain [-1, 1] and range [-p/2, p/2], and is defined by: arcsin x = y if and only if sin y = x For x in [-1, 1], sin(arcsin x) = x For x in [-p/2, p/2], arcsin(sin x) = x

    9. Arcsine function

    10. Example 1.a Find arcsin In order to do this, we need to find a number in the interval [-p/2, p/2] whose sin is . What values work? Since sin(p/6) = and p/6 is in the interval [-p/2, p/2], we have arcsin( )= p/6

    11. Example 1.b Find arcsin(sin p/3) Is p/3 in the interval [-p/2, p/2] ? Yes, so using the arcsin property, we get: arcsin (sin p/3) = p/3

    12. Example 1.c Find arcsin(sin 3p/4) Is 3p/4 in the interval [-p/2, p/2] ? No. So we need a number in the interval [-p/2, p/2] whose sine is the same as that of 3p/4. Since sin(3p/4) = v(2) / 2, what value x in the interval [-p/2, p/2] will give us sin x = v(2) / 2? sin (p/4). So we have: arcsin (sin 3p/4) = arcsin (sin p/4) = p/4.

    13. General Notes The inverses for the other trig functions are defined by making domain restrictions similar to those made for the sine function. For the cosine function, we will restrict our domain to [0, p]. Notice, the function on this interval is one-to-one and assumes all the values in the range of the ordinary cosine function. This restricted function has an inverse, namely the inverse cosine, or arccosine, function.

    14. Arccosine Function The arccosine function, denoted arccos, has domain [-1,1] and range [0, p] and is defined by: arccos x = y if and only if cos y = x For x in [-1, 1], cos (arccos x) = x For x in [0, p], arccos (cos x) = x

    15. Arccosine Function

    16. Example 2.a Find cos[arccos(- )] What is this really saying? Since the arccos (- ) is the number in [0, p] whose cosine is , the cosine of this number must be . That is: cos[arccos(- )] = -

    17. Example 2.b Find arccos(cos p/3) We need to find the number in the interval [0, p] whose cosine is the same as the cosine of p/3. Since p/3 is in the interval [0, p] we have: arccos(cos p/3) = p/3

    18. Example 2.c Find arccos [ cos (- p/4)] Is p/4 in the interval [0, p]? No. But since the cosine function is even, we know that p/4 is in the interval and that: arccos [ cos (- p/4)] = arccos [ cos (p/4)] Thus: arccos [ cos (p/4)] = p/4

    19. More to come Theres lots more in this chapterbut we need our quiz instead!!

    20. Example 2.d Lets recall what we are doing Find sin [ arccos (-5/13)] ? = arccos (5/13) ? So cos ? = 5/13 Use the Pythagorean Identity to find the third side 132 52 = x2 ? x = 12

    21. Example 2.d So we have: Sin ? = sin ( arccos (5/13)) = 12/13 Now, since sine is positive in both quadrants I and II, the sine of the arccos (-5/13) is the same as the sine of the arccos (5/13). Hence, sin ( arccos (-5/13)) = 12/13

    22. Things to note The arccosine function is not used extensively in calculus (unlike arcsine for instance). Another function that is used often is the arctangentlets explore that!

    23. Arctangent The arctangent function, denoted arctan, has domain (-8, 8) and range (-p/2, p/2), and is defined by: arctan x = y if and only if tan y = x For x in (-8, 8), tan(arctan x) = x For x in (-p/2, p/2), arctan(tan x) = x

    24. Arctangent

    25. Example 3 Find cos (arctan (12/5) + arcsin (3/5)) First note, this problem requires the sum formula for the cosine. cos (a + b) = cos a * cos b sin a sin b What is a? What is b? Here, a = arctan 12/5 and b is arcsin 3/5 Plugging these values into the formula, we get

    26. Example cos (arctan (12/5) + arcsin (3/5)) = cos (arctan (12/5)) * cos (arcsin (3/5)) sin (arctan (12/5)) * sin (arcsin (3/5)) Now, notice that sin (arcsin (3/5) is simply 3/5 Using triangles (and/or trig identities) we can solve for the rest of the values as well

    27. Example Using the triangle at the bottom, we know that ? = arctan (12/5), that is: tan ? = 12/5. Sin ? = sin (arctan (12/5)) = 12/13 Cos ? = cos (arctan (12/5) = 5/13

    28. Example Using the triangle (not to scale) at the bottom, we can also get: ? = arcsin (3/5), so sin ? = 3/5 So, we get: Cos (arctan (12/5) + arcsin (3/5)) = 5/13 * 4/5 12/13 * 3/5 = -16/65

    29. Example 4 Verify the identity: Sin (arccos x) = v(1 x2)

    30. Example 4 Using the Pythagorean identity, we get: (sin(arccos x))2 + (cos(arccos x))2 = 1 Now, cos(arccos x) = x, so: (sin(arccos x))2 + x2 = 1 And thus: sin(arccos x) = v(1-x2)

    31. Example 4 sin(arccos x) = v(1-x2) Now, the range of the arccosine function is what? [0, p]. What sign (positive or negative) does the sine function have over this interval? Positive, so we have: sin(arccos x) = v(1-x2) So we are done!

    32. Arcsecant Function The arcsecant function, denoted arcsec, has domain (-8, -1] U [1, 8) and range [0, p/2) U (p/2, p] and is defined by: arcsec x = y if and only if sec y = x For x in (-8, -1] U [1, 8), sec(arcsec x) = x For x in [0, p/2) U (p/2, p], arcsec(sec x) = x

    33. Arcsecant

    34. More notes The inverse cosecant, denoted arccsc, and inverse cotangent, denoted arccot, are defined in a similar manner. These are not used very often, and as such we will not worry about them.

    35. Homework 4.8: 1-36 by 3s, 38, 40, 42, 46

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