Circular Functions And Trig Test Answers
Circular functions and Trig test!NON CALCULATOR !!!!!!!!!!!!!!– 12 –5.(a)M13/5/MATME/SP1/ENG/TZ2/XX/Mvalid approach to find pmax min, p 6egamplitude2A1p 3(b)valid approach to find qegq(M1)period4, qN2[2 marks](M1)2periodA12N2[2 marks](c)valid approach to find rmax min, sketch of horizontal axis, f (0)egaxis2r(M1)A12N2[2 marks]Total [6 marks]6.evidence of antidifferentiationeg(6e 2t t )s 3e 2tt22(M1)A2A1CNote: Award A2 for 3e 2t , A1 fort2.2attempt to substitute (0, 10) into their integrated expression (even if C is missing) (M1)correct workingeg10 3 C , Cs 3e 2tt22(A1)77A1N6Note: Exception to the FT rule. If working shown, allow full FT on incorrectintegration which must involve a power of e.[7 marks]
5.Note: All answers must be given in terms of m. If a candidate makes an error thatmeans there is no Circularm in their answer,do notandawardTrigthe finalfunctionstestA1FT !!!!!!!!!!!!!METHOD 1(a)valid approach involving Pythagoras(M1)me.g.1sin 2 x cos 2 x 1 , labelled diagramcorrect working (may be on diagram)e.g.m2(cos100)1, 1 m(A1)21 m2cos100(b)2mtan1001 m2acceptm1 m2A1N2[3 marks]A1N1[1 mark](c)valid approach involving double angle formulae.g. sin 22sin cos2m 1 m 2sin 200accept 2m(M1)1 m2A1N2Note: If candidates find cos1001 m 2 , award full FT in parts (b) and (c),even though the values may not have appropriate signs for the angles.[2 marks]Total [6 marks]METHOD 2(a)valid approach involving tan identitysine.g. tancos(M1)correct [3 marks]– 10 –N12/5/MATME/SP1/ENG/TZ0/XX/Mcontinued Question 5 continued(b)tan100mcos100A1N1[1 mark](c)valid approach involving double angle formulame.g. sin 22sin cos , 2mtan100sin 2002m 2tan100(M1)A12m cos100N2[2 marks]Total [6 marks]6.(a)any correct equation in the form rwhere a isa tb (accept any parameter for t)54 , and b is a scalar multiple of42A2N2
Circular functions and Trig test– 10 !!!!!!!!!!!7.(a)attempt to expande.g. (sin x cos x)(sin x cos x) ; at least 3 termscorrect expansione.g. sin 2 x 2sin x cos x cos2 xf ( x) 1 sin 2 xM12/5/MATME/SP1/ENG/TZ1/XX/M(M1)A1AGN0[2 marks]A1A1N2(b)Award A1 for correct sinusoidal shape with period 2π and range [0, 2], A1 forminimum in circle.Note:[2 marks](c)p2, k2A1A1N2[2 marks]Total [6 marks]
– 11 –Circular functions and Trig !!!!!!!!!!!!M04/521/S(1)M QUESTION 9METHOD 12cos 2 x 2sin x cos x2cos 2 x 2sin x cos x 02cos x (cos x sin x) 0cos x 0, (cos x sin x) 0(M1)(M1)(A1)(A1)x ,x 24(A1)(A1)(C6)METHOD 2Graphical solutionsEITHERfor both graphs y 2cos 2 x , y sin 2 x ,(M2)ORfor the graph of y 2cos 2 x sin 2 x .(M2)THENPoints representing the solutions clearly indicated1.57, 0.785(A1)(A1)x , x 24(A1)(A1)(C6)(M1)(A1)(A1)(C3)Notes: If no working shown, award (C4) for one correct answer.Award (C2)(C2) for each correct decimal answer 1.57, 0.785.Award (C2)(C2) for each correct degree answer 90 , 45 .Penalize a total of [1 mark] for any additional answers.QUESTION 10(a)Note:(b)Note:d 40(4t 5 5e t )dt, (A1) for both limits, (A1) for 4t 5 5eAward (M1) ford 2t 2 5t 5e4tt(A1)(A1)0Award (A1) for 2t 2 5t , (A1) for 5e t . ( 32 20 5e 47 5e44)(5)(47.1, 3sf )(A1)(C3)
!!!!!!!!Circular functions and Trig test!!!CALCULATOR SECTION!– 12 –M13/5/MATME/SP2/ENG/TZ1/XX/MSECTION B8.(a)(b)evidence of choosing cosine ruleeg c 2 a 2 b 2 2ab cos C , CD 2(M1)AD22 CD AD cos Dcorrect substitutioneg 11.52 82 2 11.5 8cos104 , 196.25 184cos104A1AC 15.5(m)A1(i)METHOD 1evidence of choosing sine ruleˆsin A sin B sin ACDsin Deg,abADACcorrect substitutionˆsin ACDsin104eg815.516 ˆACD 30.0(M1)A1A1METHOD 2evidence of choosing cosine ruleeg c 2 a 2 b 2 2ab cos Ccorrect substitutioneg 82 11.52 15.516 2 2(11.5) (15.516 ) cosCˆACD30.0(ii)(c)A1A1ˆ from 73subtracting their ACDˆ , 70 30.017.eg 73 ACDN2(M1)A1correct substitution(A1)1ADC (8) (11.5)sin10422area 44.6 (m )(d)N2(M1)ˆACB43.0egN2[3 marks]N2[5 marks]areaattempt to subtracteg circle ABCD , r 2ADCACB1area ACB (15.516 ) (14)sin 42.98 ( 74.0517 )2correct working12eg44.6336(15.516 ) (14)sin 42.98 , 642shaded area is 82.4 (m2)A1N2[2 marks](M1)(A1)A144.6 74.1A1N3[4 marks]Total [14 marks]
Circular functions and Trig test!!– 15 d approacheg 13 diameter, 13 122(M1)maximum height 135 (m)A1(i)(ii)N2[2 marks]602.4A1period 25 (minutes)AGN02( 0.08 )25A1N1periodb[2 marks](c)METHOD 1valid approachegmax 74 , a(M1)135 13, 74 132a61 (accept a 61 )a61(A1)A1N2[3 marks]METHOD 2attempt to substitute valid point into equation for h2 12.5eg 135 74 a cos25(M1)correct equationeg 135 74 a cos ( ) , 13 74 a(A1)a61A1N2[3 marks]continued
– 16 –!!!!!!!!!!!!!!!Question 10 continuedM13/5/MATME/SP2/ENG/TZ1/XX/MCircular functions and Trig test(d)A1A1A1A1N4Note: Award A1 for approximately correct domain, A1 for approximately correct range,A1 for approximately correct sinusoidal shape with 2 cycles.Only if this last A1 awarded, award A1 for max/min in approximatelycorrect positions.[4 marks](e)setting up inequality (accept equation)eg h 105 , 105 74 a cos bt , sketch of graph with line y 105(M1)any two correct values for t (seen anywhere)eg t 8.371 , t 16.628 , t 33.371 , tA1A141.628,valid approach16.628 8.371 t1 t2 2 8.257 2 12.5 8.371eg,,,25255025M1p 0.330A1N2[5 marks]Total [16 marks]
Circular functions and Trig test! 4)!!!!! QUESTION 9 METHOD 1 2cos 2sin cos2 x xx (M1) 2cos 2sin cos 02 xxx 2cos (cos sin ) 0xx x (M1) cos 0,(cos sin ) 0xxx (A1)(A1) (A1)(A1) (C6)
CONCEPT IN SOLVING TRIG EQUATIONS. To solve a trig equation, transform it into one or many basic trig equations. Solving trig equations finally results in solving 4 types of basic trig equations, or similar. SOLVING BASIC TRIG EQUATIONS. There are 4 types of common basic trig equations: sin x a cos x a (a is a given number) tan x a cot x a
Solving trig inequalities finally results in solving basic trig inequalities. To transform a trig inequality into basic ones, students can use common algebraic transformations (common factor, polynomial identities ), definitions and properties of trig functions, and trig identities, the most needed. There are about 31 trig identities, among them
Derivatives of Trig Functions – We’ll give the derivatives of the trig functions in this section. Derivatives of Exponential and Logarithm Functions – In this section we will get the derivatives of the exponential and logarithm functions. Derivatives of Inverse Trig Functions – Here we will look at the derivatives of inverse trig functions.
Trigonometry (on a very basic level) trigonometric relations in right angles, values and properties of trig functions, graphing trig functions, using trig identities, solving trig equations What Not to Study Trigonometry beyond the very basics. However, you should know: SOH-CAH-TOA how to solve right triangles the unit circle
TOPIC 1.5: CIRCULAR MOTION S4P-1-19 Explain qualitatively why an object moving at constant speed in a circle is accelerating toward the centre of the circle. S4P-1-20 Discuss the centrifugal effects with respect to Newton’s laws. S4P-1-21 Draw free-body diagrams of an object moving in uniform circular motion.File Size: 1MBPage Count: 12Explore furtherChapter 01 : Circular Motion 01 Circular Motionwww.targetpublications.orgChapter 10. Uniform Circular Motionwww.stcharlesprep.orgMathematics of Circular Motion - Physics Classroomwww.physicsclassroom.comPhysics 1100: Uniform Circular Motion & Gravitywww.kpu.caSchool of Physics - Lecture 6 Circular Motionwww.physics.usyd.edu.auRecommended to you based on what's popular Feedback
Section 6.3 Inverse Trig Functions 379 Section 6.3 Inverse Trig Functions . In previous sections we have evaluated the trigonometric functions at various angles, but at times we need to know what angle would yield a specific sine, cosine, or tangent value. For this, we need inverse functions. Recall that for a one-to-one function, if . f (a) b,
TrigonometricExpressions , Equations, and Proofs . Zoom400 can simplify trig onometric expressions by using trig identities. Zoom400 can also solve equations that involve trig functions. When a trig equation has multiple answers, Zoom Math 400 will usually give only one answer. For example, entering the problem . tan . x 1. will result in the .
To use basic trig identities 3. To solve trig equations. 4. To use trig formulae (Compound, Double and Half Angle). 5. To use the form !"# % ’ ()% Key Terms or Formulas: - Refer to the Formula Sheet Assessments: - 3 marks on Short Question and 6 marks on Short Test 2
