GRAPH RESULT OF ADDING FUNCTION SIN WITH COSIN

To graph the result of adding the sine and cosine functions, we are looking at the function:

\[
f(x) = \sin(x) + \cos(x)
\]

This function combines the oscillations of sine and cosine. The graph will show periodic behavior, with the amplitude of the combined wave being \(\sqrt{2}\) (due to the trigonometric identity for the sum of sine and cosine).

Let's plot this function.The graph above shows the function \( f(x) = \sin(x) + \cos(x) \). The combined wave oscillates between approximately \( -\sqrt{2} \) and \( \sqrt{2} \), reflecting the result of adding the two periodic functions. The amplitude is enhanced compared to the individual sine or cosine waves.

Would you like more details on this graph or have any questions?

Here are 5 related questions to explore further:
1. What is the maximum and minimum value of \( \sin(x) + \cos(x) \)?
2. How can we express \( \sin(x) + \cos(x) \) using a single trigonometric function?
3. What is the period of the function \( \sin(x) + \cos(x) \)?
4. How does phase shift affect the graph of \( \sin(x) + \cos(x) \)?
5. How does the amplitude of the function \( \sin(x) + \cos(x) \) compare to the individual amplitudes of \( \sin(x) \) and \( \cos(x) \)?

**Tip:** You can rewrite \( \sin(x) + \cos(x) \) as \( \sqrt{2} \sin(x + \pi/4) \), which makes it easier to understand its amplitude and phase shift!

Explore the graph of the function f(x) = sin(x) + cos(x), which combines the periodic oscillations of sine and cosine. The amplitude of this function is √2, and its behavior oscillates between ±√2. Learn about key concepts such as phase shift, amplitude, and the sum of trigonometric functions.

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