T933263
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#1
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I don't understand any of the compound angle formulas and how to use them. I need help on this question:
sin(x+60)+cos(x+30)=0.5 between 0<x<360

Please help I'm struggling so bad.
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Sinnoh
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They simply tell you how to rewrite sines and cosines of a number in a way that might be more useful.

You've probably seen the identities \sin(A+B) \equiv \sin(A) \cos(B) + \cos(A) \sin(B) and \cos(A + B) = \cos(A) \cos(B) - \sin(A) \sin(B)

If you use that to rewrite sin(x+60) and cos(x+30) in those forms then it might be more obvious what to do next. In this case for the sine part you could have A as x and B as 60, and then for the cosine, A would be x and B would be 30.
Last edited by Sinnoh; 1 month ago
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T933263
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(Original post by Sinnoh)
They simply tell you how to rewrite sines and cosines of a number in a way that might be more useful.

You've probably seen the identities \sin(A+B) \equiv \sin(A) \cos(B) + \cos(A) \sin(B) and \cos(A + B) = \cos(A) \cos(B) - \sin(A) \sin(B)

If you use that to rewrite sin(x+60) and cos(x+30) in those forms then it might be more obvious what to do next. In this case for the sine part you could have A as x and B as 60, and then for the cosine, A would be x and B would be 30.
Is this what you mean : sin(x)cos(30)+cos(x)sin(60)=0.5 ?
I don't understand what to do next.
Is it possible for you to show me the working out?
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Sinnoh
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(Original post by T933263)
Is this what you mean : sin(x)cos(30)+cos(x)sin(60)=0.5 ?
I don't understand what to do next.
Is it possible for you to show me the working out?
You haven't applied the identity correctly.

sin(x+60), on its own, is equivalent to sin(x)cos(60) + cos(x)sin(60). Do you see how the identity gets you from sin(x+30) to that?

and then cos(x+30) becomes cos(x)cos(30) - sin(x)sin(30) if you follow the angle addition formula for cosines.
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T933263
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(Original post by Sinnoh)
You haven't applied the identity correctly.

sin(x+60), on its own, is equivalent to sin(x)cos(60) + cos(x)sin(60). Do you see how the identity gets you from sin(x+30) to that?

and then cos(x+30) becomes cos(x)cos(30) - sin(x)sin(30) if you follow the angle addition formula for cosines.
I understand that part. Thanks. But what would I do next?
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Sinnoh
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(Original post by T933263)
I understand that part. Thanks. But what would I do next?
will cos(30), sin(60), those terms, they're just constants. So you can replace them with whatever they're equal to, just to tidy them up a bit.

I think after that the next step is factorising and converting it into the form R \sin \ \text{or cos} (x + \alpha) if it doesn't simplify easily.
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Sinnoh
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Having worked through it, you don't need to do anything complicated like that. It simplifies quite nicely.
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T933263
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Thank you, I got the answer.
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