MTH112 Trigonometry 2 2 2, 2. 5π 6. cscθ = 1 sinθ = r y. secθ = 1 cosθ = r x. cotθ = 1 tanθ = cosθ. central angle time. = θ t.

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1 MTH Trigoometry,, 5, y 90 0, 5 0,, (, 0) x, 7, , ,,, Defiitios: siθ = opp. hyp. = y r cosθ = adj. hyp. = x r taθ = opp. adj. = siθ cosθ = y x cscθ = siθ = r y secθ = cosθ = r x cotθ = taθ = cosθ siθ = x y Radia: Measure of the cetral agle θ that itercepts a arc s equal i legth to the radius r. Full circle = radias. radias = 80 Arc Legth: s = rθ (θ must be i radias) r s = Agular Speed (velocity): ω = cetral agle time = θ t θ r Liear Speed (velocity): ν = arc legth time = s t = rθ t = rω

2 Graphs of Sie ad Cosie Fuctios: For the fuctios y = d + asib( x c) ad y = d + acosb( x c) Wave base lie : y = d Amplitude = a Max: y = d + a Mi: y = d a Period = b Phase Shift = c Agle: θ = b x c Graphs of Taget Fuctio: For the fuctios y = d + a tab( x c) Base lie for iflectio poit: y = d, Period = b, Vertical asymptotes for taget: b( x c) = + ; Whe the agle is straight up or dow Iverse Fuctios: Domai Rage arcsie x x y arccosie x x 0 x arctaget x x y Simple Harmoic Motio: Equatio: d = asiω t or d = acosω t where: d = distace from origi, t = time, a = amplitude, period = ω, frequecy = ω Bearigs Trigoometry: agle θ is measured couter-clockwise from the positive x-axis Ocea Navigatio ad Surveyig: Directio give as a acute agle east or west of a orth or south referece lie E.g. N W: west of due orth or θ = 90 + = 5 S 0 W : 0 west of due south or θ = = 50 (or 90 0 = 0 ) Aircraft Navigatio: Directio give as a agle measured i degrees clockwise from due orth E.g. Due orth is 0 (simpler tha 0 which would be the same directio) Due east would be a course of 90, south = 80, west = 70 Idetities f ( θ) ad v = g( θ). Note: u ad/or v may also represet fuctios of a agle: u = Pythagorea: si u + cos u = + ta u = sec u + cot u = csc u ad siu = ± cos u... etc. Cofuctio Idetities: si u = cosu cos u = siu ta u = cot u cot u = tau sec u = csc u csc u = sec u Odd Fuctios: Eve Fuctios: si( u) = siu ta( u) = tau cos( u) = cosu csc( u) = csc u cot u sec( u) = sec u = cotu

3 Chapter 7Aalytic Trigoometry Sum ad Differece Formulas: si u + v si u v tau + tav ta( u + v) = tautav = siucosv + cosusiv cos( u + v) = cosucosv siusiv = siucosv cosusiv cos( u v) = cosucosv + siusiv = ta u v tau tav + tautav Double Agle Formulas: siu = siucosu tau = tau ta u cosu = cos u si u cosu = cos u cosu = si u Half-Agle Formulas: si u = ± cosu ta u = siu + cosu = cosu siu cos u = ± + cosu Power-Reducig Formulas: si cosu u = ta cosu u = + cosu cos u = + cosu Sum to Product Formulas: siu + siv = si u + v cos u v siu siv = cos u + v si u v cosu + cosv = cos u + v cos u v cosu cosv = si u + v si u v Product to Sum Formulas: siu siv = cos u v cosu cosv = cos u v [ cos( u + v) ] siu cosv = si( u + v) + si( u v) + cos( u + v) si( u v) [ ] [ ] cosu siv = si u + v ]

4 Chapter 8: Solvig Triagles I ay triagle r ABC, with agles A, B, C, ad sides a, b, c Law of Sies: a si A = b sib = c sic Law of Cosies: a = b + c bc cos A b = a + c ac cosb c = a + b abcosc A c b B a C Area of a Triagle Area = bh I a Oblique Triagle: Area = bc si A = absic = ac sib Hero s Area Formula: Area = s( s a) ( s b) ( s c) where s = a + b + c Vectors Vector arithmetic with compoets: For u = u,u ad v = v,v u + v = u + v, u + v u v = u v, u v k u = ku,ku Magitude: Dot Product: u = ( u ) + ( u ) u v = u v + u v Agle betwee vectors: cosθ = Work: W = F D u v u v Chagig Rectagular/Polar Forms of Vectors: Rectagular: a,b, a is horizotal (x) compoet, b is the vertical (y) compoet Polar: ( r,θ), r is magitude (legth), θ agle clock-wise from positive x-axis r = a + b b θ = ta (check sigs of a, b for quadrat) a a = rcosθ b = rsiθ

5 Vector Forms of Complex Numbers Quadratic Formula For ax + bx + c = 0, x = b ± b ac a Covertig betwee rectagular ad trigoometric (polar) forms: For complex umber z: Stadard or rectagular form: Polar or trigoometric form: z = a + bi. z = r cosθ + isiθ with θ i [ 0, ) or 0,0 [ ). z = a + bi = r cosθ + isiθ r = a + b ad θ = ta b a a = r cosθ ad b = rsiθ Absolute Value (Magitude) of a Complex Number z = a + bi = r = a + b Multiplicatio of Complex Numbers i Trigoometric Form [ r ( cosθ + isiθ )] r ( cosθ + isiθ ) [ ] = r r [ cos( θ + θ ) + isi( θ + θ )] Divisio of Complex Numbers i Trigoometric Form = r r cosθ + isiθ r cosθ + isiθ [ + isi( θ θ )] r cos θ θ DeMoivre's Theorem (Powers of Complex Numbers): z = [ r( cosθ + i siθ )] = r ( cos θ + i si θ ) Roots of Complex Numbers z = r cosθ + isiθ = r cos θ + k + isi θ + k where k = 0,,,,

6 Chapter : Sequeces, Series, ad Probability. Sequeces ad Series Ifiite Sequece: A fuctio whose domai is the set of positive itegers. a,a,a,a,,a, Where the terms of the sequece are: a = f, a = f, a = f, a = f, etc. Summatio Notatio: The sum of the first terms of a sequece a i = a + a + a ++ a i= = ( ) Factorial: Fiboacci Sequece (a recursive sequece) a =, a =,..., a k = a k + a k Terms: 0,,,,, 5, 8,,,, 55, 89 Properties of Summatios: (c is a costat, a ad b are th terms of differet series). c = c. ca i = c a i i= i=. ( a i + b i ) = a i + b i. ( a i b i ) = a i i= i= i= Series: Summatio of sequeces Fiite Series ( th partial sum of the sequece) Ifiite Series (sum all terms of a ifiite sequece) i= i= i= i= b i a i = a + a + a ++ a i= a i = a + a + a ++ a i +. Arithmetic Sequeces ad Partial Sums Arithmetic Sequece: Cosecutive terms have the same differece: a a = a a = = a a = d Test: Subtract terms. There is a commo differece betwee each term: a a = d th term of the sequece: a = a + d( ) a = first term of the sequece d = commo differece = ordial umber of the term to be foud Terms: First a + d = a + d(0) = a Secod a = a + d( ) = a + d Third a = a + d( ) = a + d Fourth a = a + d( ) = a + d, etc. Note: Sice the sequece could start with ay umber, a, the formula s secod part has d( ). That meas for the first term, =, the secod part is zero leavig just a. Sum of a Fiite Arithmetic Sequece (author s formula): i= = a + d( ) S = a + a = umber of terms to sum a = first term d = commo differece Sum of a Fiite Arithmetic Sequece (alterate): S = a + d( ) = a + d( )

7 . Geometric Sequeces ad Series Geometric Sequece: Terms i the sequece have a costat ratio a = a = a = = r a a a th term i the sequece is i the form: a = a r a = first term of the sequece ( ote: a r = a r 0 = a = a ) r = commo ratio (which is the base of the expoetial) = ordial umber of the term to be foud Geometric Series: Sum of a Fiite Geometric Sequece a r i r = a r i = = umber of terms to be summed a = first term of the sequece ( a r 0 = a = a ) r = commo ratio r Sum of a Ifiite Geometric Series ( r <) a r i = a r. Coutig Priciples (Combiatorics) Fudametal Coutig Priciple: If evet E ca occur i m differet ways ad evet E ca occur i m ways after E has occurred, the total umber of ways the two evets ca occur is m m. Permutatios of Elemets: Selectig subsets of a group of items where the order of selectio matters (usig three letters, ABC is differet tha BCA). The umber of differet permutatios (differet orderigs) of thigs is. Permutatio of Elemets Take r at a Time: (Note: You are selectig r elemets ad the order selected matters.) P r = ( r) Distiguishable Permutatios: (Note: You are selectig from a pool of items, some of which are idetical. Agai, order of selectio matters.) A set of objects has k differet types of items where is the umber of oe type, the umber of the secod type, ad so o such that: = k. Distiguishable Permutatios = k Combiatios: Selectig subsets of a group of items where order does ot matter. (e.g. select three letters ABC is the same as BCA because the same letters were selected) Combiatios of Elemets Take r at a Time: C r = = ( P r r)r r i =0

Elementary Algebra and Geometry

Elementary Algebra and Geometry 1 Elemetary Algera ad Geometry 1.1 Fudametal Properties (Real Numers) a + = + a Commutative Law for Additio (a + ) + c = a + ( + c) Associative Law for Additio a + 0 = 0 + a Idetity Law for Additio a +