# Why does f[x]=Sin[2x]+Sin[3x] Not Equal a Trig Function?

• skyriver
In summary, The conversation is about the relationship between harmonic and trigonometric functions. The speaker asks for the definition of these terms and explains that a simple harmonic wave is a function with a specific form. The question is posed whether the sum of two harmonic waves with different frequencies can also be a harmonic wave.
skyriver
for example
if f[x]=Sin[2x]+Sin[3x] then f[x] can not be a Trigonometric function,
why? how many ways to prove this?

You used the word "harmonic" in your title and "trigonometric" in the text. Would you give the definition of those words that you are using?

What i mean by a simple harmonic wave is a function which has this form f(t)=ASin[ωt+φ], where A is the amplitude, ω is the frequency, and φ is the phase,t is time.

my question is :if two two harmonic waves A1*Sin(ω1*t+φ1) and B1*Sin(ω2*t+φ2) have different frequencys ,i.e ω1 is not equal to ω2, then the sum of these two functions can not be any harmonic wave ,i.e the sum can not be function like ASin(ωt+φ) .

## 1. Why is f[x]=Sin[2x]+Sin[3x] not considered a trigonometric function?

The function f[x]=Sin[2x]+Sin[3x] is not considered a trigonometric function because it cannot be expressed as a single trigonometric function such as sine, cosine, or tangent. It is a combination of two trigonometric functions, but it cannot be simplified or reduced further.

## 2. Can f[x]=Sin[2x]+Sin[3x] be written in terms of other trigonometric functions?

No, f[x]=Sin[2x]+Sin[3x] cannot be written in terms of other trigonometric functions. It is not possible to express it as a single trigonometric function or as a combination of other trigonometric functions.

## 3. Why is it important to determine if a function is a trigonometric function?

It is important to determine if a function is a trigonometric function because these functions are used extensively in mathematics, physics, and engineering. They describe periodic phenomena such as waves and oscillations and are fundamental in solving equations and analyzing data in these fields.

## 4. How can we graph f[x]=Sin[2x]+Sin[3x] without using a trigonometric function?

Since f[x]=Sin[2x]+Sin[3x] cannot be expressed as a single trigonometric function, we can still graph it by plotting points using the values of x and f[x]. This will give us a graph that shows the behavior of the function, but it will not be a smooth, continuous curve like a graph of a single trigonometric function.

## 5. Can f[x]=Sin[2x]+Sin[3x] be simplified or manipulated using trigonometric identities?

No, f[x]=Sin[2x]+Sin[3x] cannot be simplified or manipulated using basic trigonometric identities. This is because it is already in its simplest form and cannot be reduced further. However, it is possible to use other mathematical techniques, such as calculus, to analyze and manipulate this function.

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