Inverse Trig Functions
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1 Inverse Trig Functions Trig functions are not one-to-one, so we can not formally get an inverse. To efine the notion of inverse trig functions we restrict the omains to obtain one-to-one functions: [ Restrict to π, π ] Restrict to [0, π] ( Restrict to π, π The inverse trig functions are the inverses of these restricte trig functions. Notation: We use both sin x an arcsin x to represent inverse sine (similarly with the others. AMAT 7 (University of Calgary Fall 03 / 5
2 Compute sin (0, sin ( an sin (. These come from the graph: sin (0 0 sin ( π sin ( π Compute cos (0, cos ( an cos (. These come from the graph: cos (0 π cos ( 0 cos ( π Compute tan (0, lim tan x, x lim tan x. x These come from the graph: tan (0 0 lim tan x π x lim tan x π x AMAT 7 (University of Calgary Fall 03 / 5
3 Cancellation Rules The cancellation rules are tricky since we restricte the omains of the trigonometric functions in orer to obtain inverse trig functions: Cancellation Rules sin(sin x x, x [, ] sin (sin x x, x [ ] π, π cos(cos x x, x [, ] cos (cos x x, x [0, π] ( tan(tan x x, x (, tan (tan x x, x π, π Fin sin (/. Range: sin (x outputs values in [ π/, π/], thus the answer must be in this interval. Trick: Let θ sin (/. Nee to compute θ. Take sin of both sies: sin θ sin(sin (/ / by the cancellation rule. There are many angles θ that work. We want the one in the interval [ π/, π/]. Thus, θ π/6 by the special triangle an SOH CAH TOA. ( sin π 6. AMAT 7 (University of Calgary Fall 03 3 / 5
4 Harer examples Cancellation Rules sin(sin x x, x [, ] sin (sin x x, x [ ] π, π cos(cos x x, x [, ] cos (cos x x, x [0, π] ( tan(tan x x, x (, tan (tan x x, x π, π Compute cos (cos(0, cos (cos(π, cos (cos(π, cos (cos(3π Range: cos (x outputs values in [0, π], thus the answers must be in this interval. The first two we can cancel using the cancellation rules: cos (cos(0 0 cos (cos(π π The thir one we can not cancel: cos (cos(π is NOT equal to π But we know that cos(π cos(0: cos (cos(π cos (cos(0 0 Similarly with the fourth one, we can NOT cancel yet. Using cos(3π cos(3π π cos(π: cos (cos(3π cos (cos(π π AMAT 7 (University of Calgary Fall 03 4 / 5
5 Harer examples Cancellation Rules sin(sin x x, x [, ] sin (sin x x, x [ ] π, π cos(cos x x, x [, ] cos (cos x x, x [0, π] ( tan(tan x x, x (, tan (tan x x, x π, π Fin cos (cos ( 5π/3. Range: cos (x outputs values in [0, π], thus the answers must be in this interval. We can not cancel yet. Instea, we a/subtract multiples of π until we get a number in this range. By the perioicity of cos x, we have cos ( 5π 3 ( cos π 5π cos 3 ( π 3. By the cancellation rule: ( ( cos (cos 5π cos cos 3 ( π 3 π 3. AMAT 7 (University of Calgary Fall 03 5 / 5
![: Rewrite the expression cos(sin x without trig functions. (Note that the omain of this function is all x [, ]. Trick: Let θ sin x. Nee to compute cos θ.](/docs-images/50/26828725/images/6-0.png "Take sin of both sies: sin θ sin(sin (x x by the cancellation rule.")
6 : Rewrite the expression cos(sin x without trig functions. (Note that the omain of this function is all x [, ]. Trick: Let θ sin x. Nee to compute cos θ. Take sin of both sies: sin θ sin(sin (x x by the cancellation rule. Use SOH CAH TOA an a right triangle (sin is opp/hyp an sin θ x/: If z is the remaining sie, then by the Pythagorean Theorem: Since z is a length, z + z + x z x z ± x : x Thus, cos θ x by SOH CAH TOA, so, cos(sin x x AMAT 7 (University of Calgary Fall 03 6 / 5
7 Derivative of the arcsin function x sin x x Proof: Formula: [f ] (x f [f (x] Applying this formula we get: x sin x f [f (x] We let f (x sin x an f (x sin x f (sin x Substituting f (x sin x cos(sin x Since f (x cos x an f (sin x cos(sin x x Using previous example: cos(sin x x AMAT 7 (University of Calgary Fall 03 7 / 5
8 Derivatives Summary Inverse Trig Derivatives (Stanar an with chain rule x sin x x x sin (f (x (f (x f (x x cos x x x cos (f (x f (x (f (x x tan x + x x tan (f (x + (f (x f (x Fin the erivative of sin (4x. ( sin (4x (8x (4x Fin the erivative of tan (sin x. ( ( tan sin x + (sin x 8x 6x 4 x AMAT 7 (University of Calgary Fall 03 8 / 5
9 Hyperbolic Functions We efine two new functions calle hyperbolic sine an hyperbolic cosine: sinh x ex e x cosh x ex + e x Typically, sinh is pronounce sinch (or shine an cosh as you expect (rhymes with wash We can also efine hyperbolic functions for the other trigonometric functions as you woul expect: tanh x sinh x cosh x The graphs of sinh x an cosh x are: cschx sinh x sechx cosh x cothx tanh x Domain/Range: D(sinh x R R(sinh x R D(cosh x R R(cosh x [, AMAT 7 (University of Calgary Fall 03 9 / 5
10 Motivation The hyperbolic functions get their name from the fact that they have the same relationship to the hyperbola that the trigonometric functions have with the circle. No there is no traitional angle associate with hyperbolic functions (but hyperbolic angles o exist! Instea, we think of the argument as being twice the area of the sector. In the case of the circle, t can also be interprete as the raian measure of the angle. AMAT 7 (University of Calgary Fall 03 0 / 5
11 Computing particular values Compute sinh 0, cosh 0, tanh 0. Recall the efinitions: sinh x ex e x Then we have: cosh x ex + e x sinh 0 e0 e 0 cosh 0 e0 + e 0 + tanh 0 sinh 0 cosh tanh x sinh x cosh x Compute tanh(ln. tanh(ln sinh(ln cosh(ln e ln e ln e ln + e ln e ln e ln(/ e ln + e ln(/ (/ + (/ (/ + (/ 3/ 5/ 3 5 AMAT 7 (University of Calgary Fall 03 / 5
12 Hyperbolic Function Ientities Ientities cosh x sinh x sinh( x sinh x 3 cosh( x cosh x 4 cosh(x + y cosh x cosh y + sinh x sinh y 5 sinh(x + y sinh x cosh y + cosh x sinh y Some Proofs: Graphically: (cosh t, sinh t is the point on the hyperbola x y, thus, cosh t sinh t. sinh x is an o function, thus, sinh( x sinh x. 3 cosh x is an even function, thus, cosh( x cosh x. Algebraically: Using the efinitions: sinh t et e t ( e cosh x sinh x + e x x sinh( x e x e x (ex e x 3 cosh( x e x + e x cosh(x ( e x e x sinh(x cosh t et + e t ex + e x + 4 ex + e x 4 AMAT 7 (University of Calgary Fall 03 / 5
13 Derivative of the hyperbolic sine Proof: sinh x cosh x x ( e x x sinh x e x Definition of sinh x ( (ex (e x Rewriting Hyperbolic Derivatives (Stanar an with chain rule (ex ( e x Derivative of e f (x is e f (x f (x ex + e x Combine fractions cosh x Definition of cosh x sinh x cosh x x cosh x sinh x x x tanh x sech x Fin the erivative of sinh(4x. ( sinh(4x 8x cosh(4x AMAT 7 (University of Calgary Fall 03 3 / 5
14 Inverse Hyperbolic Functions Both sinh x an tanh x can be seen to be one-to-one functions so have inverses. However, cosh x is not one-to-one so we restrict the omain of cosh x to [0,. Then we efine the inverse hyperbolic cosine function as the inverse of this restricte function. Inverse Hyperbolic Functions y sinh x sinh y x y cosh x cosh y x, with y 0 y tanh x tanh y x Recall that the hyperbolic functions are efine as combinations of exponential functions. It makes sense that the inverse hyperbolic functions can be written in terms of logarithms. Inverse Hyperbolic Functions sinh x ln(x + x + cosh x ln(x + tanh x ln ( + x x with x R x with x with x (, Derivatives of Inverse Hyperbolic Functions x sinh x + x x cosh x x x tanh x x AMAT 7 (University of Calgary Fall 03 4 / 5
15 Proof of first one Prove x sinh x using the fact that + x sinh x ln(x + x +. x sinh x ( x + x + x + x + By x ln(f (x f (x f (x x + x + x + x + x + x + x + ( + (x + / (x ( + x x + ( x + + x x + By the generalize power rule Canceling s an rewriting Common enominator Canceling common factor. AMAT 7 (University of Calgary Fall 03 5 / 5
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