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.

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

4.7 Inverse Trig Functions Objectives: 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?

Precalculus 4.7 Inverse Trigonometric Functions 8 Inverse Sine Function: Arcsin ·If siny = x, then y = arcsinx ·Sometimes labeled y=sin -1 x (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 -1 x (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 -1 x (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.

Precalculus 4.7 Inverse Trigonometric Functions 12 Inverse Properties: Example ·Find the exact value (when possible)

Precalculus 4.7 Inverse Trigonometric Functions 13 Composition: Example ·Find the exact value of  x y

Precalculus 4.7 Inverse Trigonometric Functions 14 Composition: Example ·Find the exact value of  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 why the domain is [-1,1], and the range is [0, π] for the arccos function

Precalculus 4.7 Inverse Trigonometric Functions 17 Assignments