4.7 INVERSE TRIGONOMETRIC FUNCTIONS

For an inverse to exist the function MUST be one- to - one

• A function is one-to-one if for every x there is exactly one y and for every y there is exactly one x.

• So• If x and/or y is raised

to an even power then the inverse does not exist unless the domain is restricted.

• The equation y = x2

• does not have an inverse because two different x values will produce the same y-value.

• i.e. x = 2 and x = -2 will produce y = 4.

• The horizontal line test fails.

• In order to restrict the domain, a basic knowledge of the shape of the graph is crucial. This is a parabola with (0,0) as the vertex. Restrict the domain to the interval [0,infinity) to make it one-to-one.

Now let’s look at the trig functions

x

y

x

y

x

y

y = sin x y = cos x

y = tan x

x

y

For the graph of y = sin x, the Domain is (-∞, ∞) the Range is [-1, 1]

Not a 1-1 functionSo it currently does not have an inverse

x

y

However we can restrict the domain to [- Note the range will remain [-1, 1]

Now it’s 1-1!

x

y y = sinx

The inverse of sinx

or

Is denoted as arcsinx

x1sin

On the unit circle:

x

y

For the inverse sine function with angles only from -toour answers will only be in either quadrant 1 for positive values and quadrant 4 for negative values.

Find the exact value, if possible,

1 -11 3arcsin sin sin 2

2 2

x

y

y = cos x is not one to one, so its domain will also need to be restricted.

y = cos x is not one to one, so its domain will also need to be restricted.

x

y

On this interval, [0, ] the cosine function is one-to-one and we can now define the inverse cosine function.y = arccos x or y = cos-1 x

y = arccos x

y = cos x

On the unit circle ,inverse cosine will only exist in quadrant 1 if the value is positive and quadrant 2 if the value is negative.

x

y

Find the exact value for:

-12 3arccos arccos( 1) cos

2 2

y = tan x

x

y

Remember that tangent is undefined at -and

y = tanx

y = arctanx

x

y

Remember that tangent is undefined at -and

Find the exact value

1 3arctan 1 tan 0 arctan

3

Using the calculator.

• Be in radian mode• Arctan(-15.7896)• Arcsin(.3456)• Arccos(-.6897)• Arcsin(1.4535)• Arccos(-2.4534)

H Dub

• 4-7 Page 349 #1-16all, 49-67odd