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Expert: - 7/10/2009


Question
QUESTION: I know part i) but part ii) i only finished halfway because 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. Thanks a million

ANSWER: For part 2, we know that
5 sin 2A + 12 cos 2A = 11 for A between 0 and 180 degrees.

It is also known that sinČ2A + cosČ2A = 1, so it can be said
cosČ2A = 1 - sinČ2A.  This means that cos 2A = √(1 - sinČA).

This will give us a quadratic equation when it is put back into
5 sin 2A + 12 cos 2A = 11.  We get 5 sin 2A + 12 √(1 - sinČA) = 11.

Keep the square root on the left side of the equation and subtract
5 sin 2A from both sides.  This gives
12 √(1 - sinČA) = 11 - 5 sin 2A.

Squaring both sides gives
144(1 - sinČ2A) = 121 - 110 sin 2A + 25 sinČ2A.

Working this out gives
144 - 144 sinČ2A = 121 - 110 sin 2A + 25 sinČ2A.

Moving everything to the right side gives
169 sinČ2A - 110 sin 2A - 23.

Let x = sin 2A, and you have the quadratic equation
169xČ - 110x - 23 = 0.

Solve for x, know that it is sin 2A, and then angle 2A can be found.
Divide by 2 and you have A.


---------- FOLLOW-UP ----------

QUESTION: Thanks for your help. I need you help again,please.For sin 2A=.81739. I get the A=27.412degree.But I cant get the second angle, A=175.2degree.How to get the second angle? Much appreciated

Answer
To get the second angle, look at the sin() wave.
It is symmetrical around 90 degrees.
For example, the sin(89) = sin(91), sin(60)=sin(120),
sin(90-a)=sin(90+a).

For sin 2A = 0.81739, it can be found that 2A = 54.824.
This would make A = 54.824/2 = 27.412.

For the other solution, note that 2A could also be 180 - 2(54.824).
That is, 2A would also be 70.351, so A would be 70.351/2 =
35.176.

That rounds up to a 6 at the end since the 70.351 is slightly less than the true A.  I have over 8 digits of accuracy since I keep the entire numbers in Excel.

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