Hello, I have face a deadlock while trying to find the way to evaluate this identities. I'm curious that we need to use the trigonometry sum formula to find the answer of this problem, but I still cannot find the way to finish it, I always go back to the initial problem. Pls help me, so I can answer my curiosity.. Thanks

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TopNewestYou can simplify this to \(4sin 13^\circ cos 31^\circ cos 7^\circ\).

You use the following facts to simplify your expression:

\(1)\) \(sin A + sin B=2sin(\frac{A+B}{2}) cos(\frac{A-B}{2})\)

\((2)\) \(sin A - sin B=2sin(\frac{A-B}{2}) cos(\frac{A+B}{2})\)

\((3)\) \(cos A + cos B=2cos(\frac{A+B}{2}) cos(\frac{A-B}{2})\)

\((4)\) \(cos A - cos B=-2sin(\frac{A+B}{2}) sin(\frac{A-B}{2})\)

If you want to know the proofs for these, they are pretty straightforward.

For example:

We can derive, \(sin(x+y)+sin(x-y)=2sinxcosy\)

Now, let \(x+y=A\) and \(x-y=B\), and substitute in the above equation.

This will give you the proof for equation \((1)\)

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thanks Aditya, now I've just got some knowledge about simplifying the trigonometric identities..:)

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Aditya your 4th identity is wrong as it should be B-A instead of A-B which is very important as it matters a lot in sin.

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The 4th identity can, alternatively, also have a negative sign, which I supposedly missed while posting my comment. Thanks.

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Hint:\( 53 - 39 = 14 = 25 - 11\), so that suggests a place to start. Likewise, \( 53 - 25 = 28 = 39 - 11 \).Log in to reply

what is this sir...I cant understand!

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Think about how to apply the hint, esp in light of the sum and product trigonometric formulas that Aditya mentioned.

Since \(53 - 39 = 14 \), when we combine \( \cos 53 + \cos 39\), we will get \( \cos \frac{53-39}{2} \) as one of the terms in the product. Since \( 25 - 11 = 14 \), when we combine \( - ( sin 25 + \sin 11) \), we will get \( \cos \frac{25-11}{2} \) as one of the terms in the product. This allows us to factor out the \( \cos 7 \) (which you can see in Aditya's answer) and then continue with the remaining terms.

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awh!!

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answer is 1.61077

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don't use a calc...

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Hello..?? We are now in the 21st century... Use calculator or even more sophisticated computing devices like computers installed with math soft wares like mathlab, scilab and python. There are a lot of builtin functions in that three softwares,... one of which you can convert decimal to fraction with higher precision.. you type that in the console, let the software computes...

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You are missing the point. Calculators will only get you an approximate decimal answer. He wants an exact answer in trigonometric form. See Adiya's solution.

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