Chenice
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I have been doing a past paper and got onto the question:

- Express 8 sin θ − 6 cos θ in the form Rsin(θ − α), where R > 0 and 0◦ < α < 90◦
I done this to get 10sin(θ-36.9)

Then the following question asks:
Find the greatest possible value of 32 sin x − 24 cos x − (16 sin y − 12 cos y)
as the angles x and y vary.
I rewrote this as 40sin(θ-36.9)-20sin(θ-36.9)

The answer is 60, but I just don't understand how to get this... I know max of sin graph is 1, and that there is a 40 and 20 in equation but why do you add them together? So confused, I can't visualise it.
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morgan8002
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(Original post by Chenice)
I have been doing a past paper and got onto the question:

- Express 8 sin θ − 6 cos θ in the form Rsin(θ − α), where R > 0 and 0◦ < α < 90◦
I done this to get 10sin(θ-36.9)

Then the following question asks:
Find the greatest possible value of 32 sin x − 24 cos x − (16 sin y − 12 cos y)
as the angles x and y vary.
I rewrote this as 40sin(θ-36.9)-20sin(θ-36.9)

The answer is 60, but I just don't understand how to get this... I know max of sin graph is 1, and that there is a 40 and 20 in equation but why do you add them together? So confused, I can't visualise it.
There are two angles, not one. The expression should be 40\sin(\theta -36.9) -20\sin(\phi -36.9).
Maximising this, you know the maximum value of sinx is 1 and the minimum value is -1. The first term has a positive sign, so \sin(\theta-36.9) should be maximised to maximise the expression. The second term has a negative sign, so \sin(\phi-36.9) should be minimised to maximise the expression.
Hence you have 40 - 20(-1) = 40 + 20 = 60.
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