OHW_filled - Mahoney K Project Due MA 1114-Sp 320 1—...

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Unformatted text preview: Mahoney K Project Due April 19, 2011 MA 1114-Sp' 320 1—§ Name: {Z Per: 2 3 7 UFID:\_2‘_£VI_-_§n£:_ Mr. Mahoney wanted to go over the derivation of several fundamental identities in class but just does not have enough time to do that. The following H.W. Project is Due April 19, 2011 will guide you through the process of doing several of these derivations. Complete the following... Sign the Honor Statement: "On my honor, I have neither given nor received unauthorized aid in doing this assignment." BOA. Copy the Weird Sentence Below using your usual handwriting: Signature: “The quick brown fox tries to jump over a lazy dog.” Sentence: T1" (- V {C K bVO \vVi a , a Copy the Number: "9,034,761,852” Number: You may work alone, with friends. or even with tutors as long as you long as you turn in a copy of this assignment with your work on it. The Assignment must be turned in stabled with this page completed and your name on it. In this project we will derive and explore the following identities: cos(a: + 13) = cosacosfi — sin asinfi cos(a — ,8) = cos acosfi + sina sinfj’ sin(cr + i?) = sina' cosfi’ + cosarsinfi sin(r:t — i3) = sinacosf)’ — cosasinfi I n It I sm(§—9) — c059 cos (5—9) — 51n9 . . . 9 smar=smfi=fl o2=b2+c2—2bc-cosfl A B C b2=a2+cz—2ac-cosB c2=a2+b2—2ab-cosC Page 1 of 16 Mahoney Project Due April 19, 2011 MAC1114 - Spring 2011 — UF Name: Per: 2 3 7 UF ID: - The Derivation of cos(a + B). Let a: and B be arbitrary angles (that means they can be anything). The following diagram shows the unit circle with several angles, rays, terminal points, and line segments outlined. Do: In the following observations circle or underline the correct choice to make the observation true. Observe: The angle a: Q I is not 1 in standard position and its amount of rotation is labeled with the color red. Observe: The angle — B Q I is not ] in standard position and its amount of rotation is labeled with the color blue. Observe: The angle BI is ] in standard position and its amount of rotation is also labeled with the color blue. Observe: The I terminal 1 side of the angle 3 overlaps with the [ initial [@i ] side of the angle :1. Observe: The Dark Purple angl is I is not 1 in standard position. Observe: The Light Purple angle [ is @ ] in standard position. Observe: The Light Blue angle [ is [email protected] l in standard position. Observe: The Dark Purple Anglefi I is not ] opposite the Dark Green Line Segment. Observe: The Light Purple AngleG I Is not ]opposite the Light Green Line Segment. Page 2 of 16 Mahoney Project Due April 19, 2011 MAC1114 - Spring 2011 — UF Name: Per: 2 3 7 UFID: _________-___________ Do: The measurements of the Dark Purple, Light Purple, and Light Blue labeled angles are missing. These measurements can be written in terms of a and B. To some extent the colors are a hint at their values. All the angles are oriented counter clockwise. in in the angles now. If you printed this project out in black, gray, and 1, you might want to look at the .pdf version whiie working. Do: The next diagram is the same as the picture above but the labeling of the angles has been removed. Write the coordinates of the five points (four of which are terminal points) which have been labeled in this and the previous diagram with opaque (see through) greenish dots. Hint: Compare this diagram side by side with the last one to determine the coordinates. Hint: Three of the five points are terminal points for the angles in the first diagram and will have coordinates that can be written in terms of Sines and Cosines of angles in the first diagram. Hint: Two of the points have exact numerical coordinates. . . . . . 0 "‘ l l Vi Question: One of the two pomts with exact numericai coordinates is also known as the . Do: In the following third diagram, the light and dark purple angles have been separated out of the first two diagrams and from each other. Copy the values of the angles from the first diagram and the matching coordinates from the second diagram into the appropriate blanks. Page 3 of 16 Mahoney Project Due April 19, 2011 MAC1114 - Spring 2011 - UF Name: Per: 2 3 7 UF ID: - Do: Observe that each picture shows three points. Two of which are connected by a green line segment forming a triangle. Choose two different Coiored Pencils or Crowns or Markers or Highlighter or Pens and then lightly shade one of each triangle in this and the previous two diagrams. Note: in the previous two diagrams the triangles overlap so be careful and cautious. Question: Why do we know immediately that the length of the black line segments between the green opaque dots in each triangie is 1? Answer: ' e Th<7l qi/‘t on ’l‘hc l/Vli‘b CIV‘CJC Do: Label the lengths of the line segments 1 by entering the value 1 in each blue blank. Question: Why we know that the triangles are congruent? Explain. Answer:,\,[,)fl A? kWVC +WO fiJCI CIVIQ/ +hc 01‘1le in éctwccln the fame. Observe: Since the triangles for the purple angles are congruent, we now know that the light and dark green lines. I are not ]equalin length. DCOGJYll, Cox/1)) : (XL ~x,y+ (y; X)‘ . +5) . Do: Show that the length of the dark green line segment is z ¢ 0301/ 5' [ '4 gig) m (00) by applying the distance formula to the coordinates of its end points from the second diagram. Hint: The Pythagorean Identities can be used to simplify the expression in the final steps. :l (amen—l)" + (mam): 2 (”Vans—A cos(a+fil‘l"_| + 5W (‘ +13) 3J2. ~.l (04‘0”?) f—._ Length? Dark Green Line Segment Page 4 of 16 Mahoney Project Due April 19, 2011 MAC1114 - Spring 2011 — UF Name: w UFID: ____-____ E Do: Show that the length oft—he light green line segment is (C 04' K} j’ in, 0Q) 1l2—2cosorcosfi’+25inn’sinfi (lo:(vfi)/\rllh(fi)> by applying the distance formula to the coordinates of its end points from the second diagram. Hint: The binomial squared formula can be used to expand the expression inside the radical. Hint: Even and Odd Properties can be used to simplify the expression in the initial stegs. Hint: The Pythagorean Identities can be used to simplify the expression in the final Stags. (”sot ~ (”z—pf -I~ (5 ’0‘ °‘ ‘ 5'"? (‘1’)? :: ((asot—<05P )1 + W) :\E Cofbl‘lcosofl co: [5 '1‘ C05)? Jix‘ot-tle‘notrit/s + 5';th ll 9““1 (050k (05/9 fin) Sl’mKJI'n/S Length of Light Green Line Segment Do: Fill in the following table by following the instructions in the left hand column of each row. Write your work in the right hand column. Since the light and dark green line )_ A SE en}:53arenot] (Z—ZCOSOQ'i—fi) : (W 1. equal in length, it is true that... Square both sides and write the outcomes in — 2405(AT/9> : — A 4 OJ 9‘ (0 S"?! 1— ‘ thisrow. ,. __ 2 flh'kxlh/s Subtract 2 from both sides and show anyr cancellations. | a P. ’k + 3 | _ ~ 0:4 605).”) +2 SM «AS/‘75 \ Rewrite the last row A all cleaned up. Now, divide both sides by —2 and show any cancelations. _ Page 5 of 16 Ma honey Project Due April 19, 2011 MAC1114 - Spring 2011 — UF Name: Per: 2 3 7 UF ID: - Rewritethelastrow C05<°l+ fl) = (03‘51 cos/3 V S F‘gd 5|LJB all cleaned up. Derivation offhei‘mn 3:5th ef-‘Ansles Pamela If all went well, then you should have the equation cos(a + B) = cosrxcos B — sina sinfl ./ Using cos(a: + [3) to Derive Other Identities. I Do: Verify that costs: — 6) = cosa'cosfi + sina sinfi )/ by rewriting c050: — ,8) as cos[a + (—3)) and then utilizing the Cosine Sum of Angles Formula along with even and odd properties, :cpséce [3): (o:(o< +613» 4.1 9116/ 6) JIVlOLSI'l’l(~W> _, “(0:0kC0-V6— :2 Cofol (65/3 + I'v’xslnfl Derivation .bf-‘th'e cosine Difference of “8'“ WM“ (6 l l l Do: Use mag—h B) to verify that $— - cos 6— 9-) — sin 6 .\./ <0 spree): Wane “I‘M 51‘49 Z. SWIG Cains-to Sine-:Cmfflncflon’ldemltv- Page 6 of 16 Ma honey Project Due April 19, 2011 MAC1114 - Spring 2011 — UF Name: Per: 2 3 7 UF ID: - I Do: Verify that... F ...(g-(g_.))-...j (o:@<.ll_+ e) 2 Case Eu-rikgI-Laakihg-AlggatbraMammy”fuchsia: Do: Verify that by us g the Cosine to Sine Co-function Identity on the Funky Looking Algebra Identityl for Cosine in the tabl above. . O (056: (05(g‘éT—9»: W + S/jyfigfls:\o{g ~6) :3 J {n(—§-a) Sine-tin timing-Cwfiwfim Identity" ' Do: Verify that sin(a + 13) = sin a £053 + 0051! sin 3. by using both Co—Function Identities and the Cosine Difference of Angles Formula. (o:(§v(0€+fi)): (OJ {Ed .LE) 3 (as (I .A.Q<osfi ‘l" Sl'h(Lzr—‘L>S“"P LM‘Di/V W 2’— siw GNP ‘1‘ “Ma“? nermtranamesinesum 0M!) aspen-aura Page 7 of 16 Ma honey Project Due April 19, 2011 MAC1114 - Spring 2011 — UF Name: Per: 2 3 7 UF ID: - I Do: Verify that sin(a — 13) = sin 0: cos}? — cos a: sin}? by rewriting sin(a: — 3) as sin[a: + (—3)) and then utilizing the Sine Sum of Angles Formula along with even and odd properties. a’c/ sham—‘19): Sim (ac +5135) fl : :‘ma (0:594 (33 + com SlW-fi) VClA 7" 3N“ cow/3 — 6059‘5M/3 mam vitae-sineTrifle-renneaof-ifinges semis. All Triangles can be Split into Two Right Triangles Do: Refer to the three triangles below as Alex, Billy, and Chris, and fill in the blanks in the'table. C a Alex Billyr Chris The Angle C is ,Th Angle C is The Angle C is A 5 [EL [word] K lg & [word] 0 A '6 036 [word] ...That is... ‘ ...That is... ...That is... (:4— 90°[<,=,or >1 (3— 90°[<,=,or >1 0 > 90°[<,=,or >1 The longest side is C . The longest side is C . The longest side is C . The largest angle is . The largest angle is C The largest angle is C’ Page 8 of 16 Mahoney Name: Project Due April 19, 2011 Per: 2 3 7 UFID: Oh! No! MAC1114 - Spring 2011 - UF Alex, Billy, and Chris have been copied into mindless clones (They Have No Labels), Rotated by Counter Clockwise Turns, and (oh heavens) mixed all up. See instructions on next page. L; l 3‘! C My Name is... A kX ‘ My Nameis... B I [I _ CanitbeSplit?_ N0 _ .— lb l "I f My Name is... L ll) v~ '5 Can it be Split? __ M a Can it be Split? y 65’ My Name is... Can it be Split? lf| lo My Name is... Can it be Split? [JO chvif Q I My Nameis... A (CK Can it be Split? C C lo My Name is... 8 I My Can it be Split? NO ii \ J‘c l 331.1 {a My Name is... A [6% Can it be Split? >[C 5 lo Can it be Split? y 65- Page 9 of 16 Mahoney Project Due April 19, 2011 MAC1114 - Spring 2011 — UF Name: Per: 2 3 7 UF ID: - Do: Add labels back to the triangles for the corners and sides. It is best to match the colors. If you printed this project out in black, gray, and white, you might want to look at the .pdf version while working. Do: Identify which triangles are which by writing their name in the "My Name is...” section. Do: For each triangle below, Draw a vertical line from the highest corner straight down. Do: Imagine you were to cut along the lines you just drew. Indicate which triangles can be split into two distinct right triangles by answering "yes” or ”no” in the "Can it be Split?” section. You should observe that 5 of the 9 can but 4 of the 9 can not. Question: How many of the three Alex triangles can be split into two right triangles? Answer: 3 Question: How many of the three Billy triangles can be split into two right triangles? Answer: 1 Question: How many of the three Chris triangles can be split into two right triangles? Answer: 1 Question: Among all the triangles that can be split, which bottom side appears the most? Answer: : Question: is the following statement true? “Every triangle sitting on any one of its sides can be split into two right triangles by cutting along a vertical line from the opposite corner." Answer: Fq l 36 Question: Fill in the biank to make the following statement true. “Every triangle sitting on its 1 0 h Lf't side can be split into two right triangles by cutting along a vertical line from the opposite corn ” Answer: Page 10 of 16 Mahoney Project Due April 19, 2011 MAC1114 - Spring 2011 — UF Name: Per: 2 3 7 UF ID: - Deriving the Law of Sine and the Law of Cosine In this section we will derive the following equations: sinA _ sinB sin C a_b c a2 = b2 + c2 — ZbccosA .b2 = a2 4-62 —2accosB c2 = a2 +b2 — ZabcosC From the last section we saw that every triangle can be split into two right triangles as long as it is first placed on its l6 In ?a€ side. Therefore, we need only to derive the formulas for the following triangle. 7’ Do: Draw a vertical line from the C corner down to the side labeled 6, making two right triangles. Do: Label the vertical line you just made with the letter h (for height). Do: The angle C has been split. Label the left half6 and the right halfw. Do: The side c has been split. Label the left half x and the right half 3:. Do: w in the table using the Right Triangle Definitions of Sine and Cosine: . b ‘/ , h A 5111A: ‘-' cosA= —"‘ 51nB= — cosB= Cl lo b 0' Table} Do: Using the table above! complete the following table by solving for I}, y, or x. h= IOSlVlA y= [3ch h: (35mg x: a) (o3 [2 Tillie 2 Page 11 of 16 Mahoney Project Due April 19, 2011 MAC1114 - Spring 2011 — UF Name: Per: 2 3 7 UF ID: - _ Dotfll in the table using the Right Triangle Definitions of Sine and Cosine: 2c L. y u. ‘— cosfl= "' since: —— cosw= E ‘1 0i la Table 3. Sim? = I Do: Use Table 3 to verify the identity, sinC=c.h \/ o-b by using the fact that C = 9 + a) and applying the sum of angles formula for Sine. 'm< : sm(e+w\ : 51mg C0500 + .231; +11 Gila 0! ll Table- 4.. Do: Fill in the following table by following the instructions in the left hand column of each row. Write your work in the right hand column. Starting with... h = h 0 Replace each h with one of the two ‘ _ equivalent expressions for h from table 2. q S l bl B _ b 5 1 Vi A * S ‘14 Divide both sides by a - b. On each side, X S- l \'l B _ b l ,4 show which factors cancel out. 03. b _ a: b‘ Write the cleaned up version ofthe S l '4 [2 5 l n 4 V. II preceding equation. 0; Table -5 If all went well, then you should have the equation Page 12 of 16 Mahoney Project Due April 19, 2011 MAC1114 - Spring 2011 — UF Name: w UFID:____'____ SinA_sinB a _ [1' Do: Fill in the following table by following the instructions in the left hand column of each row. Write your work in the right hand column. Rewrite the left side on the right as a Sin C __ l , 5 l V3 C fraction times sin C. C _ C Use the identity from Table 4 to rewrite I K L) the sin C on the right. Show the factors = _ ' ‘— that cancel out. 1 6’- ’0 Replace the h with one of the equivalent __ 5 S l H A expressions from Table 2. _ G K . _ " 5 [HA Write the cleaned up version of the last _ ‘— equation. _ a] “liable-65 If all went well, then you should have the equation sin C sinA sin C sinB : or = c a c b W I Observe: Together, Table 5 and Table 6 imply that for every triangle with side lengths a, b, and c and I opposite angles A, B, and C that sinA _ sinB _ sinC a b c ' The law of Sines Do: m in the table using the “left" right triangle and the Pythagorean Theorem: a2: 261+ L?“ @lfi: :_Xz 9x2: qz‘ l4; Table-.7 _ Do:_F_il!_in the table using‘the "right” right triangle and the Pvthagorearl Theorem: A, L 3. >.. >_ L b2=n+y he; ~y ya: A -h Page 13 of 16 Mahoney Name: Per: Project Due April 19, 2011 2 3 7 UFID: MAC1114 - Spring 2011 - UF Do:__F_ill_ in the table using the binomial squarefl formula for the upper right box: x+y "“73 5%. KM My +y1 - Z /L fi—MY ‘7 a” l Table-9 cz~Xz—.2><y I Do: Use Table 3 to verify the identity, a-b-cosC=h2—xy by using the fact that C = S + w and applying the sum of angles formula for cosine. qb (05C : alolo.r(9+’wl 210((0J9C05W - 5"“ ._——- 6.87.” 10> —— " E F " a z :3 . hL—X _, («L—XV W —' Tahle. 11:6. Do: Fill in the following table by following the instructions in the left hand column of each row. Write your work in the right hand column. Use Table 7 to write an expression for a2. 14" +x“ Replace the x2 with an expression from Table 9. bf + Cz—QXY— y‘ Replace the h2 with an expression from Table 8. : Simplify the expression by writing 132 and 62 as the first two terms and combining any like terms. Factor a —2y out of the last two terms. Page 14 s‘—\/‘+ CKLXy-‘yl 10‘ + (2 —Ly‘ “my Vat c2“ elyfll *Xl of 16 Mahoney Project Due April 19, 2011 MAC1114 - Spring 2011 - UF Name: Per: 2 3 '1' UF ID: - I Rewrite the EXDression so that the -2 is )0 2 + C2 ’_ 2 (\/ +X ) >/ the first factor and y is the last factor. Use Table 9 to Rewrite the middle factor. bL-l-cz—ch 3-. 2. I Use Table 2to Rewrite the last factor. = lao + C “‘2. C b S 1 VI A l/lrv Table 12 If all went well, then you should have the equation a2 = b2 + c2 — ZbccosA / Except the b’s and c‘s might be switched around. Do: Fill in the following table by following the instructions in the left hand column of each row. Write your work in the right hand column. >- A Use Table 8 to write an expression for b2. b2 = l’\ + y Replace the y2 with an expression from = '42" + C 1—,. X 13‘. A X >/ Table 9. Z Replace the h2 with an expression from _ QL ’- X 14.. (L, X —- 2 X Y Table 7. _ Simplify the expression by writing a2 and 2. 2 2 c2 as the first two terms and combining = q +' C "" 2 X '- l XV any like terms. 1. 2. Factora —2x out ofthe last two terms. 5| 'l’ C 'd l X (X + Y) H 9 Rewrite the expression so that the -2 is 2- + ( L, 2 (X + \/ ) X the first factor and x is the last factor. 1, L Use Table 9 to Rewrite the middle factor. ’l" C " 2- < X ll .81 Use Table 2 to Rewrite the last factor. qz +QZ-{Z C q (053 W Table 13 If all went well, then you should have the equation 132 = a2 + c2 — ZaccosB Except the (1’5 and C’s might be switched around. Page 15 of 16 Mahoney Project Due April 19, 2011 MAC1114 - Spring 2011 - UF Name: Per: 2 3 7 UF ID: - Do: Fill in the following table by following the instructions in the left hand column of each row. Write your work in the right hand column. UseTableStowriteaformulaforcz. (:2 = )( k + AX y + Y2 Use'l'able'ltorewritexz. = q2— L‘L + ZXY + VA UseTableBtorewriteyz. = 6:2 -— L12 + 2 XY '1' El" 1,2 Simplifythe expression by writing a2 and 2 0'2- + ‘9 Z _J 2 [41 + le b2 first and combining any like terms. a2 + V v2 (1.2 my) Factor a —2 out of the last two terms. M I If you haven‘t already done so, rearrange I the terms in the parenthesis so that the 1 negative term is last. al—l— LL’iabCDIC Apply the identity from Table 10. Table 14. If all went well, then you should have the equation \/ c2 = a2 + b2 — Zabcos C. ! Observe: Together, Table 12, Table 13, and Table 14 imply that for every triangle with side lengths a, b, l and c and opposite angles A, B, and C that... a2 =b2+c2—2bccosA 7K b2 = a2 +c2 —2accosB :2 =(Jt2-l-b2 —2abcosC The law of Cos'ihe's Question: What does the last equation in The Law of Cosines reglyce to if the angle '5 90°? MW 2 qt+gz,lab<o 0".» : 6+; _. " C: Question: Why are the Law of Sines and Law of Cosines useful formulas? : 3657' 2 9 Answer 45 ‘—l ‘6’ Ci: I,7CZ+ lPo’2'lol'76. (0)10 74:.th 131-).1“ “a : 2.22M... TPagelSofll-S C 21L l 53 o 5th Sln€0° —. 1 1.5) ...
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