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Calculus/Trigonometry, double angle

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Expert: Alon Mandes - 7/26/2009


Question
I know part i) but for part ii) I dont know how to find a. i) express 5sinx+ 12cosx in the form Rsin(x + y), where R >0 & 0degree< y <90degree,giving the value of y correct 2 d.p. ii) hence solve the equation 5sin 2a+ 12cos 2a=11, giving all the solutions in the interval 0degree< a <180degree. Please help me with part ii). For part ii) I only found out (sin^-1 (11/13) )= 57.796degree. I dont know how to continue the rest in finding the value of a. Many Thanks

Answer
I'll answer for you both parts :

I) 5sinx+ 12cosx = Rsin(x+Ø). We need to find R & Ø .
We know that sin(a+b)=sinacosb+sinbcosa . therefore:
5sinx+ 12cosx = RsinxcosØ + RsinØcosx . This gives us :
RcosØ=5 & RsinØ=12 . that means tanØ=12/5=2.4 --> Ø=67.38° . /now let's find R:
R=5/cosØ=5/0.38=13 . So,
5sinx+ 12cosx = 13sin(x+67.38°) .

II) 5sin 2a+ 12cos 2a=11 --> 13sin(2a+67.38°)=11 --> sin(2a+67.38°)=11/13 -->
sin(2a+67.38°)=0.846 --> 2a+67.38°=57.79°+360° --> 2a=350.41 --> a=175.205

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Alon Mandes

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Kind of questions I can answer : Limits, Derivatives, Integration, Implicit functions, continuousity, differentiation ,Extremum problems, Lagrange multipliers, Gradients, Surface integrals, Multi variables functions ,Multi variables Integrals,Complex variables ,Complex functions, Curves, Trajectory integrals & Vector analyse,Divergence,Rotor & word problems. Kind of question I can't answer : Economics,Combinatorics,infinite series & convergence ,Statistics & Probabilities .

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1. I'm a team member of mathnerds (math site for answering questions) 2. I'm a team member in the Student's Union of the Technion, helping students who have problems in mathematics. 3. 2 years of experience as a math teacher in college. 4. I give free homework help for high school students in Mathematics & Physics. 5. I teach part time in collage the subjects : "Digital Signal Processing" , "Random Signals & Noise" , "Complex Functions".

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Hi-Tech company : GSM4VOIP ; job possition : Algorythm developer.

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M.A in Mathematics & Bs.c in Electronics.

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