Trigonometric and Polar Functions: Unit Circle, Identities, and Polar Graphs
What you must understand
Trigonometric functions connect angles to ratios on the unit circle. You need: radian measure, reference angles, signs by quadrant, periodicity, amplitude and midline for sinusoidal models, inverse trig functions and their restricted ranges, fundamental identities, and polar coordinates including conversion to rectangular form and graphing polar curves.
Traps that show up
- Radians versus degrees: most calculus-oriented problems use radians. Convert with π radians = 180°.
- Inverse trig ranges: arcsin returns [−π/2, π/2]; arccos returns [0, π]; arctan returns (−π/2, π/2). Solving sin θ = 0.5 on [0, 2π) gives two solutions, not one.
- Amplitude versus vertical shift: for y = A sin(B(x − C)) + D, amplitude is |A|, period is 2π/|B|, midline is y = D.
- Polar conversion: x = r cos θ, y = r sin θ, r² = x² + y². A negative r reflects through the origin.
- Pythagorean identity: sin²θ + cos²θ = 1 is the workhorse for simplifying expressions.
Rule of thumb
When solving a trig equation over an interval, find all solutions in one period first, then add multiples of the period and filter to the stated interval.
Worked example
Convert the polar point (r, θ) = (2, π/3) to rectangular form. x = 2 cos(π/3) = 1, y = 2 sin(π/3) = √3. The rectangular point is (1, √3).