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PLEASE HELP ME!!! 100 POINTS!!!

Express the following as a product using the sum-to-product formula:

cos(x)+2cos(7x)+cos(13x)

User Radu Topor
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1 Answer

3 votes

Answer:


4 \cos^2 \left(3x\right)\cos \left(7x\right)

Explanation:

We have been given the following trigonometric sum:


\cos(x)+2\cos(7x)+\cos(13x)

To express this sum as a product, we can use the cosine sum to product formula:


\boxed{\begin{array}{c}\underline{\sf Sum\;to\;Product\;Formula}\\\\\cos A+ \cos B=2 \cos \left((A+B)/(2)\right) \cos \left((A-B)/(2)\right)\\\\\end{array}}

Rewrite the trigonometric sum as:


\cos(13x)+\cos(7x)+\cos(7x)+\cos(x)

Apply the sum to product formula to the first two terms and the last two terms:


2 \cos \left((13x+7x)/(2)\right) \cos \left((13x-7x)/(2)\right)}+2 \cos \left((7x+x)/(2)\right) \cos \left((7x-x)/(2)\right)


2 \cos \left((20x)/(2)\right) \cos \left((6x)/(2)\right)}+2 \cos \left((8x)/(2)\right) \cos \left((6x)/(2)\right)


2 \cos \left(10x\right) \cos \left(3x\right)+2 \cos \left(4x\right) \cos \left(3x\right)

Factor out the common term 2cos(3x):


2 \cos \left(3x\right)\left[\cos \left(10x\right) + \cos \left(4x\right) \right]

Apply the sum to product formula again to the sum inside the parentheses, cos(10x) + cos(4x):


2 \cos \left(3x\right)\left[2 \cos \left((10x+4x)/(2)\right) \cos \left((10x-4x)/(2)\right)\right]


2 \cos \left(3x\right)\left[2 \cos \left((14x)/(2)\right) \cos \left((6x)/(2)\right)\right]


2 \cos \left(3x\right)\left[2 \cos \left(7x\right) \cos \left(3x\right)\right]

Remove the unnecessary parentheses:


2 \cos \left(3x\right)2 \cos \left(7x\right) \cos \left(3x\right)

Multiply the numbers:


4 \cos \left(3x\right)\cos \left(7x\right) \cos \left(3x\right)

Rewrite cos(3x)cos(3x) as cos²(3x):


4 \cos^2 \left(3x\right)\cos \left(7x\right)

Therefore, the given trigonometric sum expressed as a product is:


\large\boxed{4 \cos^2 \left(3x\right)\cos \left(7x\right)}

User Evangelia
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