Trigonometry: Unit Circle, Identities, and Sinusoidal Models
Radians, Identities, and Sinusoids
Convert with π radians = 180°. Arc length is s = rθ with θ in radians. Memorize exact coordinates at multiples of π/6, π/4, and π/3 on the unit circle; reference angles plus the CAST sign pattern let you evaluate any related angle quickly. Core identities are the Pythagorean ones (sin²θ + cos²θ = 1, plus the tan² and cot² forms), the reciprocal identities, and tanθ = sinθ/cosθ.
For y = a·sin(b(x − c)) + d, amplitude is |a|, period is 2π/|b| (or 360°/|b| in degree mode), phase shift is c, and the axis is y = d. Model real situations by extracting these four numbers from the given maximum, minimum, and timing data: amplitude is half the difference of max and min, and d is their average.
When solving equations, find the reference angle first, then generate every solution using the period. Linear equations in sine or cosine give two families of solutions per cycle; use a general solution such as θ = θ₀ + 2πn. Never divide both sides by a trigonometric factor that could equal zero - factor instead, so no solutions are discarded.
Traps: calculator in the wrong mode; writing period as 2πb; dropping the second solution family; forgetting a negative a reflects the graph; ignoring a phase shift when fitting a model.
Worked example: Solve 2 sin(3x) = 1. Then sin(3x) = 1/2, so 3x = π/6 + 2πn or 3x = 5π/6 + 2πn. Dividing by 3 gives x = π/18 + 2πn/3 or x = 5π/18 + 2πn/3.
Rule of thumb: isolate, find the reference angle, list all quadrants, then divide by the coefficient of x - including inside the period term.