7-6 The Inverse Trigonometric Functions

Objective: To find values of the inverse trigonometric functions.

The Inverse Trigonometric FunctionWhen does a function have an inverse?

It means that the function is one-to-one.

One-to-one means that every x-value is assigned no more than one y-value AND every y-value is assigned no more than one x-value.

How do you determine if a function has an inverse?

Use the horizontal line test (HLT).

Inverse Sine Function

y

2

1

1

x

y = sin x

sin x has an inverse function on this interval.

Recall that for a function to have an inverse, it must be a one-to-one function and pass the Horizontal Line Test.

f(x) = sin x does not pass the Horizontal Line Test

and must be restricted to find its inverse.

–/2 /2

The Inverse Trigonometric Function

The inverse sine function is defined byy = arcsin x if and only if sin y = x.

Angle whose sine is x

The domain of y = arcsin x is [–1, 1].

Example 1:1a. arcsin2 6

1 is the angle whose sine is .6 2

1 3b. sin2 3

3sin3 2

This is another way to write arcsin x.

The range of y = arcsin x is [–/2 , /2].

The Inverse Trigonometric Function

Unless you are instructed to use degrees, you should assume that inverse trig functions will generate outputs of real numbers (in radians).

The Graph of Inverse Sine

Finding Exact Values of sin-1x

• Let = sin-1 x.• Rewrite step 1 as sin = x.• Use the exact values in the table to find

the value of in [-/2 , /2] that satisfies sin = x.

Example

6

2

1

6sin

2

1sin

2

1sin 1

• Find the exact value of sin-1(1/2)

The other inverse trig functions are generated by using similar restrictions on the domain of the trig function. Consider the cosine function:

Inverse Cosine Function

cos x has an inverse function on this interval.

f(x) = cos x must be restricted to find its inverse.y

2

1

1

x

y = cos x

The Inverse Trigonometric Function

0

The inverse cosine function is defined byy = arccos x if and only if cos y = x.

Angle whose cosine is x

The domain of y = arccos x is [–1, 1].

Example 2: 1a.) arccos2 3

1 is the angle whose cosine is .3 2

1 3 5b.) cos2 6

35cos6 2

This is another way to write arccos x.

The range of y = arccos x is [0 , ].

The Inverse Trigonometric Function

Unless you are instructed to use degrees, you should assume that inverse trig functions will generate outputs of real numbers (in radians).

The Graph of Inverse Cosine

Inverses of Sine and Cosine

22

x

22

x

22

x

Sin(x)

Domain:

Range:

-1≤y≤1

Cos(x)

Domain:

0≤x≤¶

Range:

-1≤y≤1

Arccos(x)

Domain:

-1≤x≤1

Range:

0≤y≤¶

Arcsin(x)

Domain:

-1≤x≤1

Range:

22

x

22

y

The Inverse Trigonometric FunctionThe other trig functions require similar restrictions on their domains in order to generate an inverse.

Like the sine function, the domain of the section of the

tangent that generates the arctan is , .

2 2

The inverse tangent function is defined byy = arctan x if and only if tan y = x.

Angle whose tangent is x

Example 3: 3a.) arctan

3 6 3 is the angle whose tangent is .

6 3

1b.) tan 33 tan 3

3

This is another way to write arctan x.

The domain of y = arctan x is (-,) .

The range of y = arctan x is (–/2 , /2).

The Inverse Trigonometric Function

Unless you are instructed to use degrees, you should assume that inverse trig functions will generate outputs of real numbers (in radians).

If = x, the sinx = -1 and

Since sin (-π/2) = -1, then

Sin (-1) = (π/2)

Evaluate each expression without using a calculator.

31Tan

)1(1 Sin)1(1 Sin22

x

22

x Whose tangent is √3

Since tan π/3 = √3 then

Tan √3 = π/3

Find the following:

• Find Sin (0.8) with a calculator.

Degree mode = 53˚

Radian mode = 0.93

• Find Cos (-0.5) without a calculator.

Cos (-0.5) = x means that cos x =-0.5 between 0 and π. Thus,

Cos (-0.5) = 2π/3

-1

-1

-1

-1

Find with and without a calculator.

)3

2cos( 1

Tan

-23Hypotenuse² will be (-2)² + 3² = √13

The cos is adj/hyp = 3/√13

Rationalize Denominator = 3√13/13

√13

Calculator answer ≈ 0.83

Find the approximate value (calculator) and exact value (without a calculator)

csc(cos (-0.4))-1

-0.4 in fraction form is -2/5

Cos = adj/hyp

Opp. =√ 5² - (-2)² = √21

Csc = 1/sin = hyp/opp = 5/√21

Rationalize denominator = 5√21/21

Calculator: 1.09

-2

5

Assignment

•Page 289 #2, 4, 5 – 8, 11 – 14•Chapter 7 Test Wednesday