Phase Shift Calculator

Phase shift describes the horizontal movement of a periodic graph relative to its basic sine or cosine curve. In the form y = A sin(Bx + C) + D or y = A cos(Bx + C) + D, the signed phase shift is -C/B. A positive value moves the graph to the right, while a negative value moves it to the left. The shift is expressed in the same angular unit used by the x-axis; this calculator uses radians. The sign often causes confusion because the expression inside the function is solved for zero. Setting Bx + C = 0 gives x = -C/B, revealing where the base cycle begins after translation. For y = sin(2x - 4), B is 2 and C is -4, so the phase shift is -(-4)/2 = 2 radians to the right. For y = sin(4x + 2), the shift is -2/4 = -0.5 radians, meaning half a radian to the left. Coefficient B also controls the period. Sine and cosine normally repeat every 2π radians, but horizontal scaling changes the period to 2π divided by the absolute value of B. A larger absolute B compresses the graph into shorter cycles. A negative B reflects the graph horizontally, while the absolute value ensures the reported period remains positive. B cannot be zero for this calculation because the expression would no longer vary with x. Amplitude A and vertical shift D do not alter phase shift. The absolute value of A controls the distance from the midline to a peak, and a negative A reflects the graph across its midline. D moves the midline up or down. These coefficients are included so you can map every value from the standard function form, even though only B and C enter the horizontal-shift formula. Sine and cosine already differ by a quarter-cycle phase relationship, but the coefficient rule is identical when each function is compared with its own parent graph. Equivalent formulas can represent the same curve with different signs or constants because periodic functions repeat. Simplifying the inside expression or factoring B can make the shift more obvious: Bx + C = B(x + C/B), so the graph moves by -C/B. Phase shifts appear in wave motion, alternating current, sound, seasonal models, signal processing, and harmonic analysis. Use this calculator to check coefficient interpretation and decimal arithmetic, then inspect the signed result, direction, and period together. When your source uses degrees, convert to radians before comparing results, or convert the final radian shift by multiplying by 180/π.