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Math 1303 - Part III -Identities: cos(A + B ) = _______________________________ cos (A – B ) = _____________________________________ sin (A + B ) = _______________________________ sin( A – B ) = ______________________________________ examples: Find cos 255o . ___________________ find sin ( - 165 o ) = ____________________ Write each of the following as a single trig. function of a single angle. cos 3 cos 2 - sin 3 cos 2 = __________________________ sin cos /2 - cos sin /2 = ____________________________ sin 2 sin + cos 2 cos = _____________________________ Other identities: Double angle Identities Look at sin 2A. We can write sin 2A = sin ( A + A ) . Now what ? What about cos 2A ? By definition of the tangent function we can write similar identities for tan(A + B ) and the tan 2A . tan ( A + B ) = ____________________________ tan 2A = _______________ End of Day 17 Exam on Day 18 101 Day 19 June 21, 2002 Half-angle Identities – Use the double angle identity cos 2A = ______________________ and the Pythagorean identity ________________ To create two new identities: sin A = cos A = Both of these can also be written as sin A/2 = cos A /2 = 2) graph each of the following b) y = 4cos2 x/4 - 4 sin2 x/2 a) y = 4 sin 3x cos 3x c) y = 4sinx cos3x - 4cosx sin 3x 3) Let sin A = -3/5 with A not in quadrant III and sin B = 12/13 with B not in quadrant I. Find a) sin 2A = ____________ b) cos 2B = __________________ c) sin (A + B ) = ______________________ d) cos ( A + B ) = ______________________ e) What quadrant is A + B in ? ___________________ Why ? f) sin A / 2 = ___________________ g) cos A / 2 = ____________________ h) tan 2A = ____________________ 4) Find sin A if sin (A + B ) = 3/5 with A + B not in Quadrant II and cos B = -3/ 5, B not in quadrant II. 102 HW: page 264: 29, 33, 35, 36, 43, 47, 51, HW: 37, 39, 41, 43, 45, 47, 51, 57 page 272: 1, 5, 9, 17, 21, 23, 25, 29, 31, 33, page 278: 1, 5, 9, 15, 17, 19, 23, 33, Equations 1) Find ALL POSSIBLE solutions of the equations tan x = 0 2) Solve for if 0 < < 360o , sin2 - sin - 1 = 0 3) Solve for x if 0 < x < 2 , 4) sin 2 - cos = 0 cos 2 = -1 , find all if 0 < < 2 _ 5) \/3 sec = 2 6) If sin = 2/3 and is not in QI find a) sec = __________ b) cos 2 = ___________ b) sin 2 = _______________ sin 2 = _____________ 7) Solve. csc 3 = 2 8) cos - sin = 1 103 Another type of equation – parametric equations A parametric equation is an equation in which both x and y depend on a third variable ( a parameter ). Sometimes these are useful and sometimes they are necessary. y = t2 ex. x = 2t ex. x = sin t , y = cos t Relations - Functions – inverse relations – inverse functions Recall basic ideas about functions and relations Inverse relations: Let f(x) = 2x + 4 be given – rewrite in the form y = 2x + 4. What happens when we interchange the x and the y variables. Solve for y and label this new relation ( function ? ) g(x) = ____________ Look at the relationship between f(x) and g(x). If ( p, q) is a point on f, give me a point on g. 104 Notation: Let f(x) = 2x – 3 We can define and g(x) = 4 – x f + g: f–g: fg: f/g: Begin with a number: say 4 double it and add 3 --after you finish, take the resulting value and square it In terms of functions: Let f(x) = _______________ and g(x) = __________ (double your number and add 3 ) (square the value) Is there one function that does both functions at the “same” time – in the same equation ? Notation: We write g o f (x) and call it a composition function. g o f (x) = g ( f (x) ). ex. from above. ex. let f(x) = 3x – 2 Find g o f: find g o f ( 2 ) = __________ g(x) = x + 5 Find f o g : f o g ( - 1 ) = __________ Certain functions provide special results: ex. f(x) = 2x – 3 Find f o g ( 1 ) = _______ g(x) = ( x + 3 ) / 2 f o g ( - 2 ) = ____________ This is the idea of inverse relations. Look at the example at the previous page at the beginning of our discussion of inverse relations. This is one way of creating inverse relations. Look at trig. functions and how they relate to their inverse relations. We have solved equations of the form sin = 2/3 and found the values of that make this statement true. Change the notation of the statement above. 105 Function Domain Range f(x) = sin x f(x) = cos x f(x) = tan x f(x) = csc x f(x) = sec x f(x) = cot x all real numbers except /2, - /2 and coterminal angles all real numbers except 0, and all coterminal angles all real numbers except /2, 3/2 and all coterminal angles all real numbers except 0, and all coterminal angles Define the inverse relation of following trig. functions. What is the domain and range of each. y = sin x ________________ D: R: y = cos x _______________ D: R: y = tan x _______________ D: R: Change the range of the three relations so that they represent functions. The domain is still the same. y = arcsin x = sin-1 x D: R: y= arccos x = cos-1 x D: R: y = arctan x = tan-1 x D R: 106 Graphs: y = arcsin x y=arccos x y=arctan x Problems: Simplify each of the following. 1) sin –1 ½ = x _____________ 2) sin-1 ( - 1 ) = x __________ 3) arctan 3 ______________ 4) arc cos = x _____________ Since All of the above represent angles , we could go back and talk about the topics discussed earlier in terms of these inverse trig. functions. 5) sin ( A ) = ? If A = sin-1 4/7 ____________________ 6) cos ( arc cos 7/9 ) = _______________________ 7) sin ( tan-1 3/8 ) = ________________________ We can work with identities. 8) sin ( sin-1 3/5 + cos-1 12/13 ) = ________________________ 9) sin ( 2 arcsin 5/6 ) = _____________________ 10) cos ( 2 sin-1 (- 5/8 ) ) = _____________________________ 107 We can also solve equations. 11) 12) Oblique Triangles Law of Sines sin A/ a = sin B / b = sin C / c Can be used if one pair is known ( A, a; B, b; or C, c ) and either another angle or another side. ex. ex. Law of Cosines c2 = a2 + b2 - 2abcos C Can be used if all three sides are known or two sides and the included angle. ex. ex. 108 Examples of these type of problems. ex. 1 ex. 2 ex.3 The area of a triangle – several ways to find area - see section 7.4 on page 353. 1) (Two sides and the included angle ) ½ ab sin C or ½ ac sin B or ½ bc sin A see ex. 1 p. 354 2) ( Two angles and one opposite side ) S = a 2 sin B sin C / 2 sin A see ex. 2 p. 355 3) (Three sides ) see ex. 3 page 357 Heron’s Formula: ____________________ S = \/ s(s – a ) ( s – b) ( s – c ) , where s = ½ ( a + b + c ) 109 Vectors : magnitude and direction. Def. Vectors are said to be equal provided __________________________________________ Def. mV is a vector having the same magnitude as V but it is m times as large as vector V. if m > 0 , then it also has the same direction. if m < 0, then it has an opposite ( 180o ) direction Sum and Difference of two vectors. Def. A vector of length 1 is called a unit vector. Special vectors of length one: unit vectors in the direction of the positive x-axis and the pos. y-axis. i: ___________ j : ______________ Ai and Bj are vectors that are parallel to i and j but of length A and B respectively. Any vector can be written in terms of a horizontal and a vertical component. ex. Let V = 3i + 4j. Draw it and its vertical and horizontal components. From this drawing we can define the length of a vector by using the Pythagorean Identity. 110 Ch. 8 Complex Numbers Question Posed by Jerome Cardan – “ If someone says to you, divide 10 into two parts, one of which multiplied into the other shall produce 40, it is evident that this case or question is impossible.” With the use of complex numbers we can find the answer to the question posed above. ___ Def. We define i = \/ - 1 . with the condition that i2 = - 1 Def. A complex number is a number of the form a + bi, where a and b are any real numbers and i is defined as above. Def. In the notation a + bi 1) a is called the real part of a + bi 2) b is called the imaginary part of a + bi 3) if b = 0, then a + bi is a copy of the real number a. 4) if b 0 , then a + bi is called an imaginary number. 5) The length of a +bi is defined as we did with complex numbers The modulus of ________ a + bi = | a + bi | = \/ a2 + b2 6) The conjugate of a + bi is the number a - bi ex. Use - 3 - 4i to find 1) _______ 2)_________ 5)__________ 6)_________ 5)__________ 6)_________ 5)__________ 6)_________ ex. Use 4i to find parts 1, 2, 5, and 6 1) _______ 2)_________ ex. Use 4 + 0i to find 1, 2, 5, 6 1) _______ 2)_________ 111 Ambiguous Case. side-side-angle 112