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 \((1)\) – Aditya Parson · 3 years, 4 months ago

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– Leonardo Chandra · 3 years, 4 months ago

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

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. – Anshul Agarwal · 3 years, 4 months ago

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– Aditya Parson · 3 years, 4 months ago

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

Hint:\( 53 - 39 = 14 = 25 - 11\), so that suggests a place to start. Likewise, \( 53 - 25 = 28 = 39 - 11 \). – Calvin Lin Staff · 3 years, 4 months agoLog in to reply

– Vamsi Krishna Appili · 3 years, 4 months ago

what is this sir...I cant understand!Log in to reply

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. – Calvin Lin Staff · 3 years, 4 months ago

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awh!! – Lorraine Marie Ricafort · 1 year, 10 months ago

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answer is 1.61077 – Deepankar Gupta · 3 years, 4 months ago

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– Đức Việt Lê · 3 years, 4 months ago

don't use a calc...Log in to reply

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... – Mharfe Micaroz · 3 years, 4 months ago

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– Calvin Lin Staff · 3 years, 4 months ago

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.Log in to reply