Inverse Trig Functions and Differentiation By Dr. Julia Arnold

Since a function must pass the horizontal line test to have an inverse function, the trig functions, being periodic, have to have their domains restricted in order to pass the horizontal line test. For example: Let’s look at the graph of sin x on [-2pi, 2 pi]. Flunks the horizontal line test.

By restricting the domain from [- pi/2, pi/2] we produce a portion of the sine function which will pass the horizontal line test and go from [-1,1]. The inverse sine function is written as y = arcsin(x) which means that sin(y)=x. Thus y is an angle and x is a number. Y = sinx Y = arcsin x

We must now do this for each of the other trig functions: Y = cos x on [0, pi] Range [-1,1] Y= arccos (x) on [-1,1] range [0, pi]

arctan(x) has a range of (- pi/2, pi/2) arccot(x) has a range of (0, pi) arcsec(x) has a range of [0, pi], y pi/2 arccsc(x) has a range of [- pi/2, pi/2], y 0

Evaluate without a calculator: Step 1: Set equal to x Step 2: Rewrite as Step 3: Since the inverse is only defined in quadrants 1 & 4 for sin we are looking for an angle in the 4th quadrant whose value is -1/2. The value must be - pi/6.

Evaluate without a calculator: Step 1: Set equal to x Step 2: Rewrite as Step 3: The inverse is only defined in quadrants 1 & 2 for cos so we are looking for the angle whose value is 0. The value must be pi/2.

Evaluate without a calculator: Step 3: The inverse is only defined in quadrants 1 & 4 for tan so we are looking for the angle whose tan value is sqr(3). The tan 60 = sqr(3) or x = pi/3

Evaluate with a calculator: Check the mode setting on your calculator. Radian should be highlighted. Press 2nd function sin .3 ) Enter. The answer is .3046

Inverse Properties If -1 < x < 1 and - pi/2 < y < pi/2 then sin(arcsin x)=x and arcsin(siny)=y If - pi/2 < y < pi/2 then tan(arctan x)=x and arctan(tany)=y If -1 < x < 1 and 0 < y < pi/2 or pi/2 < y < pi then sec(arcsec x)=x and arcsec(secy)=y On the next slide we will see how these properties are applied

Inverse Properties Examples Solve for x: If - pi/2 < y < pi/2 then tan(arctan x)=x and arctan(tany)=y Thus:

Inverse Properties Examples If where 0 < y < pi/2 find cos y Solution: For this problem we use the right triangle Sin(y) = x, thus the opp side must be x and the hyp must be 1, so sin y = x 1 x By the pythagorean theorem, this makes the bottom side Cos(y) =

Inverse Properties Examples If find tan y Solution: For this problem we use the right triangle By the pythagorean theorem, this makes the opp side 1 2 tan(y) =

Derivatives of the Inverse Trig Functions

Examples Using the Derivatives of the Inverse Trig Functions Note: u’ = du/dx Find the derivative of arcsin 2x Let u = 2x du/dx = 2

Examples Using the Derivatives of the Inverse Trig Functions Note: u’ = du/dx Find the derivative of arctan 3x Let u = 3x du/dx = 3

Examples Using the Derivatives of the Inverse Trig Functions Find the derivative of Let

Some more examples: Let find Solution: Use the right triangle on the coordinate graph Now using the triangle we can find sec x after we find the hyp.

Some more examples: Write the expression in algebraic form Let then Solution: Use the right triangle Now using the triangle we can find the hyp. 3x y 1

Some more examples: Find the derivative of: Let u =

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