Inverse Trig Functions

Transcription

1 Inverse Trig Functions 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 sin x! Thus, y = sin x is the angle whose sine is x. Another way of saying this is: y = sin x is the same as sin y = x. The fact that sin an sin are inverse functions can be expresse by the following equations: sin sin a = a for a, sin sin b = b for π b π. Since the restricte sin takes angles in the range π x π an prouces numbers in the range y, sin takes numbers in the range y an prouces angles in the range π x π. / y = sin -x - - / arcsin = π 6, since sin π 6 =.

2 sin ) = π, since sin π ) =. Sine an arcsine are inverses, so they uno one another but you have to be careful! sin arcsin ) =, but arcsin sin π) = 0, not π. 5 5 arcsinstuff) can t be π, because arcsin always returns an angle in the range π x π. Fin tan sin 5 3. picture: First, let θ = sin 5 3. This means that sin θ = 5. Now sin θ = opposite 3, so I get the following hypotenuse 3 5 I got the ajacent sie using Pythagoras: 3 5 =. Using the triangle, I have tan sin 5 3 = tanθ = 5. You can fin a erivative formula for arcsin using implicit ifferentiation. Let y = arcsin x. This is equivalent to x = sin y. Differentiate implicitly: x = sin y, = cos y)y, cosy. I like to express the result in terms of x. Here s the right triangle that says x = sin y: x y - x is, I foun the other leg using Pythagoras. You can see that cos y = x. Hence, arcsin x =. x x. That

3 Every erivative formula gives rise to a corresponing antierivative formula: = arcsin x + C. x 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. You get the inverse cosine by inverting cos x, restricte to 0 x π. y = arccos x - You get the inverse tangent by inverting tanx, restricte to π < x < π. y = arctan x / - / You get the inverse secant by inverting sec x, restricte to 0 < x < π together with π < x < π. / y = arcsec x - 3

4 As with sin an sin, the omains an ranges of these functions an their inverses are swappe : Function Domain Range arcsin x π x π arccos x 0 x π arctan < x < π < x < π arcsec x, x 0 x < π, π < x π 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. arccos x = x arctan x = + x arcsec x = x x Derive the formula for sec x. The erivation starts out like the erivation for sin x. Let y = sec x, so sec y = x. Differentiating implicitly, I get sec y tan y), so There are two cases, epening on whether x or x. sec y tan y. x y x - x - -x - x x - y 4

5 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, sec y = x an tany = x. x, so x is positive, an x = x. Therefore, sec y tan y = 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 sec y = hypotenuse) must equal x. In this case, ajacent) Therefore, sec y = x an tany = x. x, so x is negative, an x = x. Therefore, sec y tan y = x x. x x = x x. This proves that x x in all cases. arcsin x + ) arcsin x = x tan x = arcsecex ) = tan x) x + arcsin x) / ) ) + 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. Hence, arctan w = w + = + w. w arctan w + arctan ) = 0. w 5

6 A function with zero erivative is constant, so arctan w + arctan w = C, a constant. But when w =, So I get the ientity C = arctan w + arctan w == arctan + arctan = π. arctan w + arctan 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 = arcsin x a + C a + x = a arctan x a + C x x a = a arcsec x a + C Derive the extene arcsin integral formula from the formula = arcsin x + C. x a x = a = x a a) a u = u u u = arcsin u + C = arcsin x a + C. u = x a, u = a, ] = a u Using the arctan formula with a =, Using the arcsin formula with a = 3, 4 + x = arctan x + C. 3 x = sin x 3 + C. + 4x = + x) = + u u = arctan u + C = arctanx) + C. 6

7 u = x, u =, = u ] x 4 + x = 0 x 4 + x 5 ) = x 4 + u u 5x = 4 5 u = x 5, u = 5x 4, = u ] 5x 4 u + u = 5 arctan u + C = 5 arctanx5 ) + C. e x = e x e x u u e = x u = e x, u = e x, = u ] e x u u = arcsin u + C = arcsin ex + C. sec x) sec x) = u tanx) u sec x) = u = tanx, u = sec x), = u ] sec x) u u = sin u + C = sin tan x + C. c 006 by Bruce Ikenaga 7