precalculus 4.7 inverse trigonometric functions 1 inverse functions ·do all functions have an...

Precalculus 4.7 Inverse Trigonometric Functions 1

Inverse functions· Do all functions have an inverse?· Only functions that are monotonic

(always increasing or decreasing) have inverses.

· In other words, only functions that are one-to-one (have no repeated y-values) have inverses.

· In other words, only functions that pass the horizontal line test have inverses.


What is an inverse function?

· Recall the inverse of the exponential function

· Logarithmic function· In general, how do we find

the inverse of a given function?

· Determine whether it has an inverse.

· Swap the x and y variables and solve for y.

da cbx

dcbxa )(log


Inverse trig functions

· Do the trig functions have inverses?· Not at first, we have to restrict the domain.· Once we restrict the domain, what are the inverses of

the trig functions?· y = sin(x) what is the x-input, y-output?· x = sin(y) swap the variables· arcsin(x) = arcsin(sin(y))· arcsin(x) = y· or sin-1(x) = y inverse trig notation· The angle whose sine is x · x is the ratio of two sides of a triangle.· arcsin(x) = y is equivalent to sin(y) = x … why?


WARNING:

· sin-1(x) does not equal 1/sin(x)

· Why?

· The -1 denotes inverse notation

· Just like f-1(x) does not denote reciprocal which would instead be · sin(x)-1 or 1/sin(x)


Try it

sin 1

2cos

3

2tan 1

4.7 Inverse Trig FunctionsObjectives:

Identify the domain and range of the inverse trigonometric functions

Use inverse trig functions to find angles

Evaluate combinations of trig functions


Calculator Practice

· Use your calculator to evaluate the three problems from earlier

· Do the calculator answers match your answers from the unit circle?

sin 1

2cos

3

2tan 1

sin 1 1

2

cos 1 3

2

tan 1 1


Inverse Sine Function: Arcsin

· If siny = x, then y = arcsinx · Sometimes labeled y=sin-1x (inverse

notation)

· -1≤x ≤1

· -π/2 ≤y ≤ π/2

· The domain is [-1,1]

· The range is [-π/2, π/2]

WHY?


Inverse Cosine Function: Arccos

· If cos(y) = x, then y = arccos(x) · Sometimes labeled y=cos-1x (inverse

notation)

· -1≤x ≤1

· 0 ≤y ≤ π

· The domain is [-1,1]

· The range is [0, π]

WHY?


Inverse Tangent Function: Arctan

· If tany = x, then y = arctanx · Sometimes labeled y=tan-1x (inverse

notation)

· -∞< x <∞

· - π/2< y <π/2

· The domain is (-∞,∞)

· The range is (- π/2, π/2)

WHY?


Example

· Find the exact value.

arccos2

2arccos( 1)

arctan0

arctan( 1)


Inverse Properties: Example

· Find the exact value (when possible)

tan(arctan( 5))

arcsin sin53

cos(cos 1 )


Composition: Example

· Find the exact value of

tan arccos2

3

x

y


Composition: Example

· Find the exact value of

cos arcsin 3

5

x

y


Evaluating inverse trig functions with a calculator

· By definition, the values of inverse functions are always in radians.

· arctan(-8.45)· sin-1(0.2447)· arccos(2)· sec-1(2)· arccsc(1)· cot-1(3)


Closure

Explain whythe domain is [-1,1], and

the range is [0, π]

for the arccos function


Assignments

precalculus 4.7 inverse trigonometric functions 1 inverse functions ·do all functions have an...

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