Sinusoidal Function
General sine wave with amplitude A, period 2π/B, phase C, shift D.
Formula
Variables
How it works
This formula describes a sine wave that has been stretched, shifted sideways, and moved up or down from the basic graph of y = sin(x). The factor A controls how far the wave rises above and falls below its center line, B changes how quickly the wave repeats, C shifts the graph left or right, and D sets the midline of the wave. It works because sine is already a repeating periodic pattern, and these transformations let us match many real oscillations without changing the underlying sinusoidal shape.
When to use it
Use this formula whenever a situation repeats in a smooth wave-like way, such as sound waves, tides, alternating current, or seasonal patterns. It is also useful for modelling data when you know the maximum, minimum, period, and vertical offset of a cycle.
Worked example, step by step
Find key features of y = 3sin(2x - π) + 1. Step 1: Identify A, B, C, and D: A = 3, B = 2, C = -π, D = 1. Step 2: Amplitude = |A| = |3| = 3. Step 3: Period = 2π / B = 2π / 2 = π. Step 4: Phase shift: rewrite 2x - π as 2(x - π/2), so the graph shifts right by π/2. Step 5: Vertical shift = D = 1, so the midline is y = 1. Step 6: Therefore the wave oscillates between 1 - 3 = -2 and 1 + 3 = 4. Final answer: amplitude 3, period π, shift right π/2, midline y = 1, maximum 4, minimum -2.
Common mistakes to avoid
- Using B as the amplitude instead of the frequency factor
- Forgetting to divide 2π by B when finding the period
- Getting the phase shift sign wrong when rewriting Bx + C in factored form
Did You Know?
Nearly every wave in nature — sound, light, tides, AC electricity — can be described by a sinusoid.
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More in Trigonometry
View allSine Ratio (SOH)
BasicRatio of the side opposite an angle to the hypotenuse.
Cosine Ratio (CAH)
BasicRatio of the side adjacent to an angle to the hypotenuse.
Tangent Ratio (TOA)
BasicRatio of the opposite side to the adjacent side.
Cosecant
IntermediateReciprocal of sine.
Secant
IntermediateReciprocal of cosine.
Cotangent
IntermediateReciprocal of tangent.