University:
Washington State University
Course:
MATH 108 | Trigonometry
Academic year:

2023
Views:

171

Pages:

8
Author:

NeatGold

INVERSE TRIGONOMETRIC FUNCTIONSsin" cos'x, tan'x and so on are inversefunctions of sinx, cosx, tanx and so on.anglessin'y =xsinx =valuesysin x is also written as arcsinxnote The domain of sin x is the range of sinz.arcsin arccos arctan ancoosec arcsec arccotdomain [-1,1] [-1,1] R R-(-1,1) Rrange [-%, 1/2] [0,"]] (o, iinote When x is +veall ITF in 1st Quadwhen x is -vesin', tan wsec- in 4thcos', cot", sec-' in 2ndalso while plotting graphs, remember thatsinx < x < tanxy tanx yz xy = sinx4545 15 Graphs of ITFarcsinxy=xarcsinx51/2sinxt-IlT1/224 > X2arcsinzy >arcinzarccosxarccasey= = xTn/2coset-1124 > x2arcosx < arcusearctanxy=nn/2E- n/224 > X2arctanx >arctanz ITFProperties flf (x)) restricted = of x only domain if of x belongs the ITF f. toFor the example, arcin(sinx) = x only whenx E [-m/2,This means that -"1/2]arcsin (sinx) = x En/2r-xx-2n+24TY2n/2It351/2r/2note-arccos (cosx) = x G [o, n}r-xx2n -x-2ntxF2nnote COST (cosx) = x)) = 2hn I x arctan (tanx) . x 6 RTVn+2THAN2n+xThn/2II3n/2-Nenote tan -1 -(tanx) = tant(taning + x) = hr + xf(f (x)) = x with no strings attachedRelations in ITFNote that we consider x>0 in all thebelow relations, to ensure the compatibilityof the functionssin'xx>0If sin'x = 0, O must belong to [- 1 1/2, T1/2]x = sinocoso = # 11 x x 2= + 1-x2 (because Q E [o, 7/2])so six''x = cost (1-22)Similarly, sin'x = tan -I x V1-x2and sin'x = cosec 1/xand sin'x = sec-' V/T-x2and sintx = cot VI-X%/cos-'x= sin 1-x2= tan V1-x= cusec-'n= sec-11/x= cot x/51xxtantx = sin X-VI+X2 = need If x to <0, be then made some to the relations- sin = -cin'xsin"=- costri-x2+ tan X/V1-x==+ cosec 1/x=-sec 1/V1-x2=+cot - 1-2% ncos' x los+ (-x) = n cos'x= n- sin -1 1-x2= + tan 11-xx/x (n= n cosec - 1/41-x2= + sec-'1/x= + cot * //VI-x2tan x - tant (-x) = - tantx= + sin-1/1/xx2= = cos-'/rltx2= + cot 1/n -rvoteran cosec - x->cosec ((-x) =- cosectsec-'xsec -1 (-x) = F - sectwt xcot ( - x) = r - wtxnote sin n + co5'x = n/2 x E - 1,1tantx + cot'x = M/2 X E Rsec + cosec = r/2 1x1 = 1alsotan xy su <1tan'x + tanty=n+ xy?n/2 xy = /tan * - tant'y = tan (2-4/1+xy)tantx + tanty + tantz = tan (x+ytz-xy's(x,y, 3 00 & (xy + 43+zx < 1) & (xy, ye zx <1) 1-(xy+y3+x) andsin + x2+++sin'x + sinty =n- sin- ( 1. )x'+y2>17/2x2+y2=also cos * + cos'y = cos' (xy-ra-x2)(1-yes) -Multiple Angles in ITFsin (2xr1-x2)n - sin'x1/52exel=2sintx-1/V2 S X E 1/5/2-F -sintx --1.6x<-1/42312I1/2E-1/2o1/2I-1/2-13/2cos' (2x- - 1)2wstx=2r - cost1 x x < 0B2SIo tan (20/1-x2)-r + 2tan"x > 12tan+-15x51=n+ 2tantxx c - 1n/2tE-1I-1N/27.sin (2x/1+x2) .r- 2tant'x x > 1=2tan'xr - 2tantx x c - 1cost(1-x-1+x+)2tan-x x30=2Ean x tosin (32-4x) -r- - 3sin'x1/2 - 2 5 1=3sin *-1/2 S x s 1/2-n- - 3sin'x-1 x x - 1/2 cost(423-3x)3cos'x=2r-3cost --2n + 3cos'x-15x<-1/2tan- (3x-x3/1-3xt)-n + 3tan'xx x 1/r3=3tantx1/5 S x 5 1/53lrt 3tantxx - 1/53 -

Inverse Trigonometric Functions

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