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Inverse Trigonometric Functions

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Definition of the Inverse Function

• Let f and g be two functions such that f (g(x)) = x for every x in the domain of g and g(f (x)) = x for every x in the domain of f.

• The function g is the inverse of the function f, and denoted by f -1 (read “f-inverse”).

• Thus, f ( f -1(x)) = x and f -1( f (x)) = x.

• The domain of f is equal to the range of f -1, and vice versa.

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Text Example

Show that each function is the inverse of the other: f (x) = 5x and g(x) = x/5.

Solution To show that f and g are inverses of each other, we must show that f (g(x)) = x and g( f (x)) = x. We begin with f (g(x)).

f (x) = 5x

f (g(x)) = 5g(x) = 5(x/5) = x.

Next, we find g(f (x)).

g(x) = 5/x

g(f (x)) = f (x)/5 = 5x/5 = x.

Notice how f -1 undoes the change produced by f.

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Finding the Inverse of a Function

The equation for the inverse of a function f can be found as follows:

1. Replace f (x) by y in the equation for f (x).2. Interchange x and y.3. Solve for y. If this equation does not define y as a

function of x, the function f does not have an inverse function and this procedure ends. If this equation does define y as a function of x, the function f has an inverse function.

4. If f has an inverse function, replace y in step 3 with f -1(x). We can verify our result by showing that f ( f -1(x)) = x and f -1( f (x)) = x.

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Find the inverse of f (x) = 7x – 5.Solution Step 1 Replace f (x) by y.

y = 7x – 5

Step 2 Interchange x and y.

x = 7y – 5 This is the inverse function.

Step 3 Solve for y.x + 5 = 7y Add 5 to both sides.

x + 5 = y7

Divide both sides by 7.

Step 4 Replace y by f -1(x).

x + 57

f -1(x) = Rename the function f -1(x).

Text Example

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The Horizontal Line Test For Inverse Functions

A function f has an inverse that is a function, f –1, if there is no horizontal line that intersects the graph of the function f at more than one point.

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The inverse sine function, denoted by sin-1, is the inverse of the restricted sine function y = sin x, - /2 < x < / 2. Thus,y = sin-1 x means sin y = x,where - /2 < y < /2 and –1 < x < 1. We read y = sin-1 x as “ y equals the inverse sine at x.”

y

-1

1

/2x

- /2

y = sin x

- /2 < x < /2

Domain: [- /2, /2]

Range: [-1, 1]

The Inverse Sine Function

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The Inverse Sine Function

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Finding Exact Values of sin-1x

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Example

6

2

1

6sin

2

1sin

2

1sin 1

Find the exact value of sin-1(1/2)

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The Inverse Cosine Function

The inverse cosine function, denoted by cos-1, is the

inverse of the restricted cosine function y = cos x, 0 < x < .

Thus, y = cos-1 x means cos y = x, where 0 < y < and –1 < x < 1.

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Find the exact value of cos-1 (-3 /2) Solution Step 1 Let = cos-1 x. Thus = cos-1 (-3 /2)

We must find the angle , 0 < < , whose cosine equals -3 /2

Step 2 Rewrite = cos-1 x as cos = x. We obtain cos = (-3 /2)

Step 3 Use the exact values in the table to find the value of in [0, ] that satisfies cos = x. The table on the previous slide shows that the only angle in the interval [0, ] that satisfies cos = (-3 /2) is 5/6. Thus, = 5/6

Text Example

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The Inverse Tangent Function

The inverse tangent function, denoted by tan-1, is the

inverse of the restricted tangent function y = tan x, -/2 < x < /2. Thus, y = tan-1 x means tan y = x,where - /2 < y < /2 and – < x < .

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Inverse Properties

The Sine Function and Its Inversesin (sin-1 x) = x for every x in the interval [-1, 1].sin-1(sin x) = x for every x in the interval [-/2,/2].

The Cosine Function and Its Inversecos (cos-1 x) = x for every x in the interval [-1, 1]. cos-1(cos x) = x for every x in the interval [0, ].

The Tangent Function and Its Inversetan (tan-1 x) = x for every real number x tan-1(tan x) = x for every x in the interval (-/2,/2).