Linear combination of sine and cosine function

In summary: I thought I must have been doing something wrong.In summary, the minimum of $6\sin x+8\cos x+5$ is $-5$. The minimum of $(6\sin x+8\cos x)^2+5,\,(6\sin x+8\cos x)^3+5$ is $0+5=5$, and the minimum of $(6\sin x+8\cos x)^4+5$ is $0+5=5$.
  • #1
anemone
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MHB
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Hi MHB! I recently came across a problem and I was thinking most likely I was missing something very obvious because I couldn't make sense of what was being asked, and I so wish to know what exactly that I failed to relate.

Question:
Find the minimum of $6\sin x+8\cos x+5$. Hence, find the minimum of $(6\sin x+8\cos x)^2+5,\,(6\sin x+8\cos x)^3+5$ and $(6\sin x+8\cos x)^4+5$.

It is important to stress that students are expected to solve it via trigonometry route but not other methods.

I would feel the "hence" implies that the first part of the question greatly help to reach to the answers for the subsequent parts of the problem, but not that I could see it...therefore, any insight would be helpful and thanks in advanced for the help!
 
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  • #2
If f(x) has minimum m at x= x0 then f(x)+ 5 has minimum m+ 5 at x0, f(x)^2+ 5 has minimum m^2+ 5 at x= x0, and, generally, f(x)^n+ 5 has minimum m^n+ 5 at x= x-x0.
 
  • #3
Country Boy said:
If f(x) has minimum m at x= x0 then f(x)+ 5 has minimum m+ 5 at x0, f(x)^2+ 5 has minimum m^2+ 5 at x= x0, and, generally, f(x)^n+ 5 has minimum m^n+ 5 at x= x-x0.
That is not true. For example, the function $f(x) = \sin x$ has minimum $m = -1$ at $x_0 = \frac{3\pi}2$. But $f(x)^2 + 5$ does not have minimum $m^2+5 = 6$ at $x = x_0$. Instead, it has minimum $5$ at $x = 0$.
 
  • #4
Argh!(Headbang) I couldn't believe how I overlooked something so trivially simple in order to deduce the minimum of the other functions! Once we rewrite $6\sin x+8\cos x+5=10\sin \left(x+\tan^{-1}\dfrac{4}{3}\right)+5$, we know the minimum can be attained at $-5$, and so the minimum values of

$(6\sin x+8\cos x)^2+5=\left(10\sin \left(x+\tan^{-1}\dfrac{4}{3}\right)\right)^2+5$ is $0+5=5$,

$(6\sin x+8\cos x)^3+5=\left(10\sin \left(x+\tan^{-1}\dfrac{4}{3}\right)\right)^3+5$ is $(-10)^3+5=-995$,

$(6\sin x+8\cos x)^4+5=\left(10\sin \left(x+\tan^{-1}\dfrac{4}{3}\right)\right)^4+5$ is $0+5=5$.

Sorry for asking something so simple!
 

Related to Linear combination of sine and cosine function

1. What is a linear combination of sine and cosine functions?

A linear combination of sine and cosine functions is a mathematical expression that combines a sine function and a cosine function with coefficients in a linear fashion. This means that the coefficients are multiplied by the sine and cosine functions, and then added together. The result is a new function that can be graphed and analyzed.

2. Why do we use linear combinations of sine and cosine functions?

Linear combinations of sine and cosine functions are useful in many areas of science and engineering, particularly in fields such as signal processing, acoustics, and electrical engineering. They allow us to model and analyze complex periodic phenomena, such as sound waves and electrical signals, in a more efficient and accurate way.

3. How do we find the coefficients in a linear combination of sine and cosine functions?

The coefficients in a linear combination of sine and cosine functions can be found using a mathematical technique called Fourier analysis. This involves using integrals and trigonometric identities to calculate the coefficients that best fit the given data or function.

4. What is the difference between a linear combination of sine and cosine functions and a Fourier series?

A linear combination of sine and cosine functions is a specific type of Fourier series, which is a mathematical tool used to represent periodic functions as a sum of sine and cosine functions. The difference lies in the specific coefficients used and the type of function being represented.

5. Can a linear combination of sine and cosine functions accurately represent any periodic function?

Yes, a linear combination of sine and cosine functions can accurately represent any periodic function. This is due to the properties of Fourier series, which state that any periodic function can be represented as a sum of sine and cosine functions with appropriate coefficients.

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