Trig Expression simplification

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SUMMARY

The discussion focuses on simplifying the trigonometric expression $\cos(3x) + 4h \cos(x)$, where $h$ is a constant. The key identity used for simplification is the triple-angle formula for cosine, specifically $\cos(3x) = 4\cos^3(x) - 3\cos(x)$. By substituting this identity into the expression, it is simplified to $\cos(x)(4\cos^2(x) - 3 + 4h)$. The participants clarify that the sum-to-product identity is not applicable due to differing coefficients on the cosine terms.

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  • Familiarity with factoring expressions in algebra.
  • Basic knowledge of trigonometric functions and their properties.
  • Ability to manipulate algebraic expressions involving constants.
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  • Study the derivation and applications of the triple-angle formula for cosine.
  • Explore factoring techniques for polynomial expressions in trigonometry.
  • Learn about sum-to-product identities and their limitations in specific scenarios.
  • Practice simplifying complex trigonometric expressions with varying coefficients.
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Students and educators in mathematics, particularly those focusing on trigonometry, as well as anyone looking to enhance their skills in simplifying trigonometric expressions.

cbarker1
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What identity I need to use for simplifying this trig expression into one expression?$cos(3x)+4hcos(x)$ where h is a constant.

Thank you for your help.

Can you explain, too?
 
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To simplify an expression that contains two terms means we need to combine these two terms to become one single term, i.e. by factoring out the common factor.

In our case ($\cos3x+4h \cos x$), we need to express $\cos 3x$ in terms of $\cos x$ since the second term has a $\cos x$ in it.

By using the triple-angle formula for $ \cos 3x$, where $ \cos 3x=4\cos^3x-3 \cos x$, we can simplify the original expression as follows:

$\displaystyle \cos3x+4h \cos x =(4\cos^3x-3 \cos x)+4h \cos x=\cos x(4\cos^2x-3+4h)$
 
I wished to use sum-product identity.
 
With differing coefficients on the cosine terms, I don't see how you can use a sum-to-product identity.
 

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