Inverse Trig Functions -7-08 If you restrict fx) = sinx to the interval π x π, the function increases: y = sin x - / / This implies that the function is one-to-one, an hence it has an inverse. The inverse is calle the inverse sine or arcsine function, an is enote arcsinx or sin x). Note that in the secon case sin x) oes not mean sinx! Note: Arcsine an arcsin x) are oler terms, an there is similar terminology for the other inverse trig functions so arctangent an arctan x for the inverse tangent function, an so on). I ll use the inverse function terminology instea. In wor, y = sin x is the angle whose sine is x. Another way of saying this is: y = sin x is the same as siny = x. The fact that sin an sin are inverse functions can be expresse by the following equations: sinsin a = a for a, sin sinb = b for π b π. Since the restricte sin takes angles in the range π x π an proces numbers in the range y, sin takes numbers in the range y an proces angles in the range π x π. / y = sin -x - - /
sin an sin ). sin = π 6, since sin π 6 =. sin ) = π, since sin π ) =. Sine an arcsine are inverses, so they uno one another but you have to be careful! sin arcsin ) =, but arcsinsinπ) = 0, not π. 5 5 sin stuff) can t be π, because sin always returns an angle in the range π x π. Example. Fin tansin 5 3. First, let θ = sin 5 3. This means that sinθ = 5 opposite. Now sinθ =, so I get the following 3 hypotenuse picture: 3 5 I got the ajacent sie using Pythagoras: 3 5 =. Using the triangle, I have tansin 5 3 = tanθ = 5. You can fin a erivative formula for sin using implicit ifferentiation. Let y = sin x. This is equivalent to x = sin y. Differentiate implicitly: x = siny = cosy)y cosy I like to express the result in terms of x. Here s the right triangle that says x = siny: x y - x
is, I foun the other leg using Pythagoras. You can see that cosy = x. Hence, y =. That x sin x =. x Every erivative formula gives rise to a corresponing antierivative formula: x = sin x+c. Before I o some calculus examples, I want to mention some of the other inverse trig functions. I ll iscuss the inverse cosine, inverse tangent, an inverse secant functions. a) You get the inverse cosine by inverting cosx, restricte to 0 x π. y = arccos x - b) You get the inverse tangent by inverting tanx, restricte to π < x < π. y = arctan x / - / 3
c) You get the inverse secant by inverting secx, restricte to 0 < x < π together with π < x < π. / y = arcsec x - As with sin an sin, the omains an ranges of these functions an their inverses are swappe : Function Domain Range sin x π x π cos x 0 x π tan < x < π < x < π sec x,x 0 x < π, π < x π tan an cos ). tan = π 4, since tan π 4 =. cos ) = π 3, since cos π 3 =. You can erive the erivative formulas for the other inverse trig functions using implicit ifferentiation, just as I i for the inverse sine function. cos x = x tan x = +x sec x = For example, I ll erive the formula for sec x. 4 x x
The erivation starts out like the erivation for sin x. Let y = sec x, so secy = x. Differentiating implicitly, I get secytany) secytany There are two cases, epening on whether x or x. x y x - x - -x - x x - y Suppose x. Then y = sec x is in the interval 0 y < π, as illustrate in the first iagram above. You can see from the picture that Therefore, secy = x an tany = x. x, so x is positive, an x = x. Therefore, secytany = x x. x x = x x. Now suppose that x. Then y = sec π x is in the interval < y π, as illustrate in the secon iagram above. Since x is negative, the hypotenuse must be x, since it must be positive an since secy = hypotenuse) must equal x. In this case, ajacent) Therefore, secy = x an tany = x. x, so x is negative, an x = x. Therefore, secytany = x x. x x = x x. This proves that y = x x in all cases. : 5
a) sin ) x+ sin x. b) tan x. c) sec e x ). a) b) c) sin ) x+ sin x = x x + sin x ) / ) ) tan x = tan x) +x. sec e x e x ) = e x e x = e x. I on t nee absolute values in the last example, because e x is always positive. x. Example. Prove the ientity tan w+tan w = π. Hence, A function with zero erivative is constant, so tan w = w + = +w. w tan w +tan ) = 0. w tan w+tan w = C, a constant. But when w =, Therefore, C = tan w+tan w = tan +tan = π. tan w+tan w = π. Here are the integration formulas for some of the inverse trig functions. I m giving extene versions of the formulas with a replacing the that you get if you just reverse the erivative formulas in orer to save you a little time in oing problems. a x = x sin a +C a +x = a tan x a +C 6
x x a = a sec x a +C For instance, here s how to erive the extene sin integral formula from the formula sin x+c using substitution: a x = a = x a a) a = u [u = xa, = a, = a ] x = u = sin u+c = sin x a +C. 4+x an 3 x. Using the tan formula with a =, 4+x = tan x +C. Using the sin formula with a = 3, 3 x = sin x 3 +C. +4x. +4x = +x) = +u = tan u+c = tan x)+c. [ u = x, =, = ] x 4 +x 0. x 4 +x 0 = x 4 +x 5 ) = x 4 +u 5x 4 = 5 ] [ u = x 5, = 5x 4, = 5x 4 5 tan u+c = 5 tan x 5 )+C. +u = e x e x. 7
e x = e x e x u e x = [ u = e x, = e x, = e x u = sin u+c = sin e x +C. ] secx) tanx). secx) secx) = tanx) u secx) = [ u = tanx, = secx), = ] secx) u = sin u+c = sin tanx+c. c 08 by Bruce Ikenaga 8