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# Help on trigonometry question

Nigelmca 22 March 2012 14:59 Posts5
Joined22/03/12

Hi All,

I am brushing up on AS Level maths and cannot agree with a text book answer.

Question.

Find the least positive value of the angle x.
sin (5x + y) = cos ( y – 3x)

Answer : I use sin z = cos ( 90 - z ) so z = 5x + y and 90 - z = y - 3x.
Simplifying 5x + y = 90 - y + 3x infers x + y = 45.
I deduce x is the smallest positive value.

Nigel.
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stevencarrwork 22 March 2012 15:31 Posts680
Joined28/07/09
sin is not a one-to-one function so sin A = sin B does not imply A = B

11.25 looks right to me.
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stevencarrwork 22 March 2012 15:51 - Last edited by stevencarrwork on 22 March 2012 15:56 Posts680
Joined28/07/09

sin(5x +y ) = sin(90-y+3x)

sin(5x +y ) - sin(90-y+3x) = 0

2 sin (x+y-45)cos(4x+45) = 0

So cos (4x+45) = 0, so 4x + 45  = 90, x =11.25

Needs checking....
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Nigelmca 23 March 2012 07:46 Posts5
Joined22/03/12
Thank you for your help Steven.

I am unsure why you have not considered the other root in the equation

2 sin (x+y-45)cos(4x+45) = 0

Sin(x+y-45) can also take a value of zero to solve the equation.

So Sin(x+y-45) = 0, so x+y-45 = 0, so x+y =45. Thus x could take a value of 1 degree and y of 44 degrees.

Back to original equation
sin (5x + y) = cos ( y – 3x)
sin((5(1)+44) = cos (44-3(1))
sin 49 = cos41. This is true
0.7547... = 0.7547 .

Anyway I contacted the publishers, OCR, who informed me that the question was probably a misprint and should have been

sin (5x + y) = cos ( 3x – y)

This definitely results in x = 11.25.

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stevencarrwork 23 March 2012 07:56 Posts680
Joined28/07/09
I assumed it had to be true for all values of y, or else the question would make no sense.

So I ignored the other solution.

But Cos (3x-y) is identically equal to cos (y-3x), so how can that second problem have a different answer?

That makes no sense.

It is like claiming the solution of x^2 + 2x = 9 is a misprint for   2x + x^2 = 9 , which has a different solution.of.

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stevencarrwork 23 March 2012 07:58 Posts680
Joined28/07/09
Your second question :- sin (5x + y) = cos ( 3x – y)

'This definitely results in x = 11.25.'

If we take x as 1 and y as 44, we get sin 49 = cos (-41)

Which is true.

So the 'solution' is x = 1, just like before.
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Nigelmca 24 March 2012 08:15 Posts5
Joined22/03/12
A few further points to consider.........

The text book has not introduced the AS level student to compound angle or double angle identities where this question is presented, so these solutions are 'out of scope' in this context.

However your comment 'sin is not a one-to-one function so sin A = sin B does not imply A = B' has led me to the solution through another route.

Solution:
sin(5x +y ) = sin(90-y+3x)
Either 5x+y = 90-y+3x          or            5x+y = 180-(90-y+3x)
x+y=45                                                    x = 11.25

I suppose the answer on the left is discarded because the solution involves multiple values of x and y.

I would prefer to have seen the question reworded as : " Find the least positive value of the angle x which holds true for any value of angle y for which  (5x + y) = cos ( y – 3x).

Anyway, thanks very much for your help, Steven. I will now move onto the next chapter !

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Name withheld 17 July 2018 06:37
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