PHYS 1114, Lecture 1, Jnury 18 Contents: 1 Discussed Syllus (four pges). The syllus is the most importnt document. You should purchse the ExpertTA Access Code nd the L Mnul soon! 2 Reviewed Alger nd Strted Trigonometry. Nine pges were shown (pges 1 9 in this file).. The functions sine, cosine nd tngent will very soon ecome essentil. The rules of sines nd cosines were lso discussed, ut once we introduce vectors nd their components we cn ypss tht. Logrithms pper lter only riefly with sound. c. Announced Mth Proficiency Test for the Fridy Clss. Try to nswer the ten questions to the est of your current ility. If you do not perform well there, you should refresh or lern the mteril s soon s possile. You will get full credit for this quiz (this only for this one, even if you get ll nswers wrong). d. You should hve simple scientific clcultor nd ring it to ll tests nd l clsses. The ility to use it for trigonometry clcultions will e tested in the Mth Proficiency Test. The use of grphics clcultors, personl digitl ssistnts (ipds etc.), cell phones, ooks nd notes of ny kind will not e llowed t the four exms. You should ring pens or pencils, nd simple scientific clcultor. (Some scientific clcultors without the ility to disply ny preprogrmmed equtions re mentioned in the syllus.) i
Bsic Alger Formule Direct nd Inverse Proportionlity x / y () x/y = k = Constnt x / 1/y () xy = k = Constnt Liner Eqution x + = 0 () x = Simultneous Equtions x + y + c = 0 dx + ey + f = 0 () etc. y = x+c = (x + c)/ dx + ey + f = 0 Qudrtic Eqution x 2 + x + c = 0 () x = ± p 2 4c 2 Powers 1 n = 1/n = n m n = m+n n m = n / m = n n n = () n m 1
( n ) m = mn x m x n = x m+n p = 2 p = 1/2 = 1 2 np = 1/n = 1 n 1/n n = 1 = 0 = 1 Common Logrithm y = 10 x () x = log y Nturl Logrithm e = 2.71828 = lim 1 + 1 n n!1 n y = e x () x = ln y log = log + log log = log(/) = log log log n = n log 2
In More Detil: Remrk : The product of nd cn e three wys, nmely = = Liner eqution in one vrile : To solve x + 5 = 4x + 3 move terms with nd without x to opposite sides x 4x = 3 5 =) 3x = 2 =) x = 2 3 = 2 3 Two liner equtions in two vriles : To solve 2x + y = 1 x + 3y = 4 you cn proceed two wys: One wy is to dd or sutrct equtions s shown: 2x + y = 1 2x + y = 1 1 2x + 6y = 8 x + 3y = 4 2 =) =) y = 7 5y = 7 5 = 7 5 Alterntively solve x (or y) from one eqution nd sustitute the result in the other eqution s follows: x = 4 3y, 2(4 3y) + y = 1. Then 8 6y + y = 1 =) 5y = 7 =) y = 7 5. Both wys we finish the sme: x = 4 3 7 5 =) x = 20 5 3 21 5 = 1 5.
Qudrtic eqution in one vrile : x 2 + x + c = 0 =) x = ± p 2 4c 2 It is proved y completing the squre : x + 2 = x 2 + 2 2 x + x 2 + x + c = 2 = 0 + 2 4c (2) 2 =) x + p 2 = ± 2 Product of powers : + 2 4c (2) 2 4c 2 m n = m+n 3 2 = = 5 = 3+2 Power of powers : ( m ) n = mn ( 3 ) 2 = 3 3 = 6 = 2 3 ( 3 ) 4 = 3 3 3 3 = 4 3 = 12 Inverse of powers : m = 1/ m = 1 m, 0 = 1 (if 6= 0) 1 3 = 3, 3 3 = 3 3 = 1 = 0 = 3 3 4
Some Specil Angles nd Degrees versus Rdins: 90º 180º 270º 360º = 0º One gets the sme ngle when dding or sustrcting 360. Note: Sometimes 1 degree gets divided in 60 rc minutes nd 1 rc minute in 60 rc seconds: 1 = 60 0 = 3600 00, 1 0 = 60 00. For rottionl motion (nd in clculus) it is etter to mesure ngles in rdins (rd): circumference = 2pr rc length = qr r r q r re = pr 2 re = - qr 2 1 2 2 rd = 360, rd = 180 1 rd = 180 degrees = 57.295 5
Definition of sin, cos, nd tn: (x nd y cn e positive, zero, or negtive, s 0 pple pple 360.) y O r θ x sin = y r, cos = x r, tn = y x = sin cos To get x nd y from r nd : x = r cos, y = r sin To get r nd from x nd y: r = p x 2 + y 2, = rctn y x = tn 1 (y/x) 6
Min definitions of trigonometry : Consider the right tringle c θ with hypotenuse of length c nd ngle, s in the figure, is the length of the short side djcent to nd the length of the opposite side. The one defines the sine, cosine nd tngent y the rtios sin = c, cos = c, tn =. Sometimes one uses the crostic sohchto to memorize this: sine = opposite djcent, cosine = hypotenuse hypotenuse, tngent = opposite djcent. Here is the Greek letter thet for th. In science one lso uses Greek letters s our Ltin lphet hs only 26 smll nd 26 cpitl letters, which is often not enough. 7
Bsic Trigonometry Formule c θ sin = /c, cos = /c, tn = / (cot = /, sec = c/, csc = c/) β c α γ c 2 = 2 + 2 2 cos / sin = / sin = c/ sin 8
Rules of Sines nd Cosines: It is possile to find the sine nd cosine rules directly from the definitions of sin nd cos, just drwing one perpendiculr: c c 2 β c 1 h α Sine rule: Cosine rule: h = sin = sin or sin = sin c 1 = cos nd c 2 = p 2 h 2 (Pythgors) =) c = c 1 + c 2 =) (c c 1 ) 2 = c 2 2 =) (c cos ) 2 = 2 2 sin 2 = 2 2 (1 cos 2 ) (Here sin 2 +cos 2 = 1 (Pythgors) hs een used.) =) c 2 2c cos + 2 cos 2 = 2 2 + 2 cos 2 =) 2 = 2 + c 2 2c cos 9