Graph of y = sin(x) + cos(x)
Interactive graph of y = sin(x) + cos(x). Explore the combined trigonometric wave, its amplitude and phase shift.
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Understanding the Function
The function y = sin(x) + cos(x) combines the two fundamental trigonometric functions into a single wave. Using the identity a*sin(x) + b*cos(x) = R*sin(x + phi), this expression can be rewritten as sqrt(2)*sin(x + pi/4), revealing that it is a sine wave with amplitude sqrt(2) (approximately 1.414) and a phase shift of pi/4 to the left.
This combination demonstrates the principle of superposition, which is fundamental in physics. When two waves of the same frequency are added, the result is always another wave of the same frequency but potentially different amplitude and phase. This principle underlies wave interference, signal processing, and AC circuit analysis.
Key properties: amplitude sqrt(2), period 2*pi, maximum value sqrt(2) at x = pi/4 + 2n*pi, minimum value -sqrt(2) at x = 5*pi/4 + 2n*pi, zeros at x = 3*pi/4 + n*pi. The derivative is cos(x) - sin(x).