[PDF] Inverse Trigonometric Functions

The Inverse Trigonometric Functions

arccos is defined in a range of [0, π] and domain of [ 1,1]− Find the exact value of the expression whenever it is defined − − 2 1 sin 1 − 2 2 cos 1 tan −1(−1) 2 1 arcsin − 2 3 arccos arctan (3) For the following problems: Always work from the inside out 2 2 sin (1)]arcsin cos[cos −1 tan [tan −1(5)] You will notice when

Inverse trigonometric functions (Sect 76) Review

y = arctan(x) y y = arccsc(x)-1 0 1 p / 2 - p / 2 x y = arcsec(x)-1 10 p / 2 p y x y 0 p / 2 x y = arccot(x) Review: Definitions and properties Theorem For all x ∈ [−1,1] the following identities hold, arccos(x)+arccos(−x) = π, arccos(x)+arcsin(x) = π 2 Proof: arccos(-x) q 1 y-x = cos (p-q) x = cos (q) p - q q x arccos(x) arccos(x) q

63: Inverse Trigonometric Functions

Other notations for inverse functions:sin 1(x) = arcsin(x), cos 1(x) = arccos(x) and tan 1 (x) = arctan(x) Note that the inverse function notation is only true for the restricted

CALCULUS Inverse functions

arccos, arctan, arccot EXERCISE Sin Sin : [ — 2 sin —00, oo Sin in x un arccos arcsin angles complementary ïFWfns yield Inve ext subt arccot R (O, T)

46 Inverse Trigonometric Functions

(a)arccos(p 2 2) (b)arccos(p 3 2) (c)arcsin(p 3 2) (d)arctan(1) (e)arctan(p 3) (f)arctan(p 3) Solutions (a)Since arccos(x) is the inverse function of cos(x) then we seek here an angle whose cosine is p 2 2:Since cos(ˇ 4) = p 2 2 then arccos(p 2 2) should be ˇ 4: (b)arccos( 5 p 3 2) = ˇ 6 since cos(5ˇ 6) = p 3 2: (c)arcsin(p 3 2) = ˇ 3

Inverse Trigonometric Functions

y arcsin sin 1x: y cosx: y arccos x cos 1 x: y xtanx: y arctan x tan 1: Trig function Restricted domain Inverse trig function Principle value range 2 2 S S d x d y arcsinx 2 2 S S d y d y cosx 0 xd S y Sarccosx 0 yd y tanx 2 2 S S x y arctanx 2 2 S S y 1 2 S 2 S 1 2 S 1 1 2 S 2 S S 2 S 2 S y x

Inverse trigonometric functions (Sect 76)

The derivative of arcsin is given by arcsin0(x) = 1 √ 1−x2 Proof: For x ∈ [−1,1] holds arcsin0(x) = 1 sin0 arcsin(x) = 1 cos arcsin(x) For x ∈ [−1,1] we get arcsin(x) = y ∈ hπ 2, π 2 i, and the cosine is positive in that interval, then cos(y) = + q 1−sin2(y), hence arcsin0(x) = 1 q 1−sin2 arcsin(x) ⇒ arcsin 0(x) = 1 √ 1

MATH 150 { TOPIC 15 INVERSE TRIGONOMETRIC FUNCTIONS I

and/or by plotting some points, we see that arcsin(x) has the following graph arcsin(x)orsin−1(x) 1 −1 ˇ 2 −ˇ 2 Fig 15 3 The derivations of the graphs of arccos(x) and arctan(x) are similar to that of arcsin(x): For arccos(x); we rst consider cosx where 0 x ˇ: cosx,0 x ˇ −1 1 ˇ 2 ˇ Fig 15 4

Some Worked Problems on Inverse Trig Functions

1 arcsin[sin(ˇ 8)]: 2 arccos[sin(ˇ 8)]: 3 cos[arcsin(1 3)]: Solutions 1 Since arcsin is the inverse function of sine then arcsin[sin(ˇ 8)] = ˇ 8: 2 If is the angle ˇ 8 then the sine of is the cosine of the complementary angle ˇ 2 ˇ 8, which, after getting a common denominator, simpli es to 3ˇ 8 In other words, the sine of ˇ 8 is the

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www.rit.edu/asc Page 1 of 7 I. Four Facts About Functions and Their Inverse Functions:

1. A function must be one-to-one (any horizontal line intersects it at

most once) in order to have an inverse function.

2. The graph of an inverse function is the reflection of the original

function about the line xy 3. If ),(yx is a point on the graph of the original function, then ),(xy is a point on the graph of the inverse function. interchanged. II. Illustration of the Four Facts for the Cosine Function:

Background: The regular cosine function for

fx , is not one-to- one since some horizontal lines intersect the graph many times. (See how the horizontal line 1y intersects the portion of the cosine function graphed below in 3 places.) Therefore more than one x value is associated with a single value. The inverse relationship would not be a function as it would not pass the vertical line test.

Inverse Trigonometric

Functions

xyyx1yxycos22xy1cos www.rit.edu/asc Page 2 of 7 FACT #1: A function must be one-to-one (any horizontal line intersects it at most once) in order to have an inverse function.

The restricted cosine function,

xycos on the interval ddx0 is one-to- one and does have an inverse function called xarccos or x1cos . See the graphs of the restricted cosine function and its inverse function below: FACT #2: The graph of an inverse function is the reflection of the original function about the line xy

Note the symmetry of graphs of

xcos and xarccos about the line xy

FACT #3: If

),(yx is a point on the graph of the original function, then ),(xy is a point on the graph of the inverse function. 2 1,3 is a point on the graph of xycos 2 1 3cosS 3,2 1 is a point on the graph of xyarccos 32

1arccos

In general, if

yxarccos , then yxcos yx1cos implies xycos xy12 2

1xycosxy1cosyx

www.rit.edu/asc Page 3 of 7 interchanged. xcos xarccos

Domain

ddx0 (restricted domain) 11dx Range 11dy ddy0 (restricted range)

Example: Evaluate

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