# Why Does Asin(ax)+Bcos(ax) Create a Sinusoidal Wave?

• mtanti
In summary, when rewriting the expression Asin(ax)+Bcos(ax) into a sinusoidal form, you can use the arbitrary constants C and d to represent the coefficients A and B. By expanding the right hand side, you can find the values for R and b, which allows you to rewrite the expression as a single trig function. This demonstrates that the sum of trigonometric functions will always result in another sinusoidal wave.
mtanti
Can someone tell me why Asin(ax)+Bcos(ax) always gives another sinusoidal wave?

Because you can rewrite that expression into a sinusoidal form.

and can that answer be translated into a more mathematical reason?

Try it out...
put A = C*cos(d) B = C*sin(d) where C and d are arbitrary constants. Of course you can put sin in the place of cos and vice-versa, and still get a sinusoidal wave.

$$A\sin ax + B \cos ax = R \sin (ax+b)$$

Expanding the right hand side gives you $$R\sin ax \cos b + R \cos ax \sin b$$

This gives $$R \cos b = A$$ and $$R \sin b = B$$, therefore $$b = \tan^{-1}\frac{B}{A}$$.

A similar method gives you R in terms of A and B, thus turning your sum of trig functions into another, single, trig function. :)

## 1. Why does asin(ax)+bcos(ax) create a sinusoidal wave?

This equation is a combination of two trigonometric functions, sine and cosine, which are known to produce sinusoidal waves. When plotted on a graph, the result is a smooth, repetitive wave pattern.

## 2. What do the a and b values represent in this equation?

The a value represents the frequency of the wave, while the b value represents the amplitude. The frequency determines how many cycles of the wave occur per unit of time, while the amplitude determines the height of the wave.

## 3. How does changing the values of a and b affect the sinusoidal wave?

Changing the value of a will alter the frequency of the wave, resulting in more or less cycles per unit of time. Changing the value of b will change the amplitude of the wave, making it taller or shorter. Both values can also affect the phase shift of the wave, which is the horizontal displacement of the wave on the graph.

## 4. Can this equation be used to model real-world phenomena?

Yes, this equation can be used to model many different types of periodic or cyclical phenomena, such as sound waves, electromagnetic waves, and oscillating systems. It is a fundamental equation in understanding the behavior of waves in various fields of science.

## 5. Are there any limitations to using this equation to model waves?

This equation is a simplified representation of waves and may not accurately model all types of waves. It also assumes that the amplitude and frequency remain constant, which may not always be the case in real-world scenarios. In addition, the equation does not account for factors such as damping or interference, which can affect the behavior of waves.

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