Combining sine and cosine into one wave

Combining sine and cosine into a single sinusoidal wave

Learn to rewrite a linear combination of sine and cosine with the same angular frequency as one wave. This precalculus set develops amplitude and phase through coefficient matching, quadrant checks, zero-coefficient cases and equivalent sine or cosine representations.

Progressive single-select and multiselect tasks connect symbolic identities with exact input–output samples, attainable ranges, equation solving and the location of a maximum. Detailed explanations distinguish phase angles from horizontal shifts and show why simply adding the two coefficients does not give the combined amplitude.

Before you begin

Review sum-and-difference identities, unit-circle signs and principal inverse tangent. Expand a proposed single-wave form to verify its coefficients. When solving an equation, check whether the requested output lies within the wave’s range before finding angles. Each item is independent, and both special cases and ordinary nonzero waves are included. These are practice tasks for mathematical modelling, not claims about measured physical data.