Trigonometry starts as triangle measurement and becomes the mathematics of anything that repeats. The unit circle is the bridge between those two views, so time spent there pays back everywhere.
Practise trigonometrySine, cosine and tangent are ratios of sides relative to a chosen angle, which is why they depend on shape but not size. Any two similar right triangles share the same ratios.
For an unknown side use the ratio directly; for an unknown angle use the inverse function. Label the hypotenuse first, then the opposite side relative to your angle, and the correct ratio becomes obvious.
On a circle of radius one, the coordinates of a point at angle θ are exactly (cos θ, sin θ). Negative and obtuse angles then need no special treatment, the coordinates simply change sign.
Reference angles reduce every angle to the first quadrant. Compute the value there, then apply the quadrant sign rule to finish.
In y = A·sin(Bx − C) + D the amplitude is |A|, the period is 2π/B, C/B is the phase shift, and D is the vertical shift. Reading these four numbers is enough to sketch the curve.
Tangent behaves differently: its period is π and it has vertical asymptotes wherever cosine is zero, which is why it has no amplitude.
Proving an identity means transforming one side into the other. Convert everything to sines and cosines, combine over a common denominator, and look for the Pythagorean identity.
Trigonometric equations have infinitely many solutions, so respect the stated interval. Find the reference solution, use symmetry for the second one in the interval, then add multiples of the period if a general solution is required.
