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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.
Precalculus 4.7 Inverse Trigonometric Functions 2
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
Precalculus 4.7 Inverse Trigonometric Functions 3
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?
Precalculus 4.7 Inverse Trigonometric Functions 4
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)
Precalculus 4.7 Inverse Trigonometric Functions 5
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
Precalculus 4.7 Inverse Trigonometric Functions 7
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
Precalculus 4.7 Inverse Trigonometric Functions 8
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?
Precalculus 4.7 Inverse Trigonometric Functions 9
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?
Precalculus 4.7 Inverse Trigonometric Functions 10
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?
Precalculus 4.7 Inverse Trigonometric Functions 11
Example
· Find the exact value.
arccos2
2arccos( 1)
arctan0
arctan( 1)
Precalculus 4.7 Inverse Trigonometric Functions 12
Inverse Properties: Example
· Find the exact value (when possible)
tan(arctan( 5))
arcsin sin53
cos(cos 1 )
Precalculus 4.7 Inverse Trigonometric Functions 13
Composition: Example
· Find the exact value of
tan arccos2
3
x
y
Precalculus 4.7 Inverse Trigonometric Functions 14
Composition: Example
· Find the exact value of
cos arcsin 3
5
x
y
Precalculus 4.7 Inverse Trigonometric Functions 15
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)
Precalculus 4.7 Inverse Trigonometric Functions 16
Closure
Explain whythe domain is [-1,1], and
the range is [0, π]
for the arccos function
Precalculus 4.7 Inverse Trigonometric Functions 17
Assignments