Notes 6 ngle Mesure Definition of Rdin If circle of rdius is drwn with the vertex of n ngle Mesure: t its center, then the mesure of this ngle in rdins (revited rd) is the length of the rc tht sutends the ngle 80 Reltionship etween 80 rd rd rd 80 Degrees & Rdins: To convert degrees to rdins, multiply y 80 80 To convert rdins to degrees, multiply y Length of irculr rc: In circle of rdius r, the length s of n rc tht sutends centrl ngle of θ rdins is: s rθ re of irculr Sector: In circle of rdius r, the re of sector with centrl ngle of θ rdins is r θ Liner & ngulr Speed: Suppose point moves long circle of rdius r nd the ry from the center of the circle to the point trverses θ rdins in time t Let e s rθ the distnce the point trvels in time t Then the speed of the oject is given s y: ngulr Speed ω θ Liner Speed v t t Reltionship etween If point moves long circle of rdius r with ngulr Liner & ngulr speed: speed ω, then its liner speedv is given yv rω Stndrd Position: oterminl (T): n ngle hs its vertex t the origin nd initil side on the positive x-xis two ngles in stndrd position re coterminl if their sides coincide Exmple : Find the 00 7 c 50 rd mesure of the ngle Exmple : Find the degree mesure of the rd 7 9 c
Exmple : Find n ngle 00 70 c etween 0 & 60 or 0 & tht is ct with the given ngle 7 d 5 Exmple 4: centrl ngle θ in circle of rdius 5m is sutended y n rc length of 6m Find the mesure of θ in degrees & rdins Exmple 5: ceiling fn With 6-in ldes rottes t 45rpm Find the ngulr speed of the fn in rd/min Find the liner speed of the tips of the ldes in in/min
Notes 6 ngle Mesure Trigonometric Rtios Specil Tringles: cscθ 0 secθ 90 60 tnθ cotθ 45 90 45 Vlues of Trigonometric rtios for Specil ngles θ in degrees θ in rdins sin θ cos θ tn θ csc θ sec θ 0 6 45 4 cot θ 60 Exmple : Sketch tringle tht hs cute ngleθ, nd find the other five trigonometric rtios ofθ : csc θ Exmple : Show tht the height h of the mountin in the figure is given y: tn β tnα d h d tn β tnα cot α cot β h d α β Use the formul in prt () to find the height h of the mountin if α 5, β 9, nd d 800ft
Notes 6 Trigonometric Functions of ngles Def of Trig Function: letθ e n ngle in stndrd position & let ( x y ) P, e point on the terminl side If r x + y is the distnce from the origin to the point P ( x, y ) then: y x y r r x,, tnθ x 0, cscθ y 0, secθ x 0, cotθ y 0 r r x y x y Signs: Trig Functions Sine ll ( ) ( ) ( ) ( ) Tngent osine Fundmentl Identities: Reciprocl Identities: csc( θ), secθ, tnθ, cotθ tnθ Pythgoren Identities: sin θ + cos θ, tn θ + sec θ, + cot θ csc θ Reference ngle: re of Tringle: Letθ e n ngle in stndrd position The reference ngleθ ssocited with is the cute ngle formed y the terminl side ofθ & the x-xis The re Α of tringle with sides of lengths nd & with included ngleθ is Α Exmple : Find θ 5 c 00 d 5 Exmple : Find the sec 0 exct vlue of the trig function 5 sin c csc 660 Exmple : Find the re of n equilterl tringle with side of length 0
Notes 64 & 65 Lw of Sines nd Lw of osines Lw of Sines: In Tringle we hve: sin sin sin c The miguous se: Two sides nd the ngle osite on e of those sides (SS) Lw of osines: Heron s Formul: In ny tringle, we hve: + c c cos + c c cos c + cos The re Α of tringle is given y: s s s s c s + + c Α ( )( )( ) where ( ) semiperimeter of the tringle; tht is, s is hlf the perimeter is the Exmple : Use the, 0, c 50 0, c 45, 5 Lw of Sines to solve for ll possile tringles tht stisfy the given conditions c 5, c 0, 5 Exmple : Use the 5, c 8, 08,? Lw of osines to find the missing side or ngleθ 5, 60, c 546,?