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# AS/A Level Further Mathematics Forum

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# Help on trigonometry question

 Nigelmca 22 March 2012 14:59 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 : .... stevencarrwork 22 March 2012 15:31 sin is not a one-to-one function so sin A = sin B does not imply A = B What number was your answer? I couldn't see your answer in your posting. 11.25 looks right to me. 17 July 2018 06:37 This post is currently under review as part of our moderation process stevencarrwork 22 March 2012 15:51 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.... Nigelmca 23 March 2012 07:46 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 eq.... stevencarrwork 23 March 2012 07:56 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 se.... stevencarrwork 23 March 2012 07:58 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 .... Nigelmca 24 March 2012 08:15 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....

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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