Trigonometric Identities

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

Note by Leonardo Chandra
6 years, 4 months ago

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5 votes

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You can simplify this to 4sin13cos31cos74sin 13^\circ cos 31^\circ cos 7^\circ.

You use the following facts to simplify your expression:

1)1) sinA+sinB=2sin(A+B2)cos(AB2)sin A + sin B=2sin(\frac{A+B}{2}) cos(\frac{A-B}{2})

(2)(2) sinAsinB=2sin(AB2)cos(A+B2)sin A - sin B=2sin(\frac{A-B}{2}) cos(\frac{A+B}{2})

(3)(3) cosA+cosB=2cos(A+B2)cos(AB2)cos A + cos B=2cos(\frac{A+B}{2}) cos(\frac{A-B}{2})

(4)(4) cosAcosB=2sin(A+B2)sin(AB2)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(xy)=2sinxcosysin(x+y)+sin(x-y)=2sinxcosy

Now, let x+y=Ax+y=A and xy=Bx-y=B, and substitute in the above equation.

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

Aditya Parson - 6 years, 4 months ago

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

Leonardo Chandra - 6 years, 4 months ago

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Hint: 5339=14=2511 53 - 39 = 14 = 25 - 11, so that suggests a place to start. Likewise, 5325=28=3911 53 - 25 = 28 = 39 - 11 .

Calvin Lin Staff - 6 years, 4 months ago

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what is this sir...I cant understand!

Vamsi Krishna Appili - 6 years, 4 months ago

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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 5339=1453 - 39 = 14 , when we combine cos53+cos39 \cos 53 + \cos 39, we will get cos53392 \cos \frac{53-39}{2} as one of the terms in the product. Since 2511=14 25 - 11 = 14 , when we combine (sin25+sin11) - ( sin 25 + \sin 11) , we will get cos25112 \cos \frac{25-11}{2} as one of the terms in the product. This allows us to factor out the cos7 \cos 7 (which you can see in Aditya's answer) and then continue with the remaining terms.

Calvin Lin Staff - 6 years, 4 months ago

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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.

ANSHUL AGARWAL - 6 years, 4 months ago

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

Aditya Parson - 6 years, 4 months ago

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awh!!

Lorraine Marie Ricafort - 4 years, 10 months ago

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answer is 1.61077

Deepankar Gupta - 6 years, 4 months ago

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don't use a calc...

Đức Việt Lê - 6 years, 4 months ago

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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...

Mharfe Micaroz - 6 years, 4 months ago

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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.

Calvin Lin Staff - 6 years, 4 months ago

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