Solving Trigonometric Equations

Calculus Readiness · Axiom Academy

LESSON Solving Trigonometric Equations Finding every angle in — and then every angle, period — by reading the unit circle. 1. One Equation, Two Solutions To solve , ask the circle a question: where is the height ? Draw the horizontal line . It slices the unit circle at two points — one in the first quadrant, one in the second — and those are your two base solutions. The same line crosses the sine wave at exactly the same two angles. 2. Every Angle, Period: the General Solution Sine repeats every , so each base solution is really a whole family . Slide the window left or right by one full turn and the two crossings reappear, unchanged. Watch the two base solutions spawn copies marching out by each — that endless ladder is the general solution . — the Q1 solution and all its -shifts. — the Q2 solution and all its -shifts. — any integer, positive or negative. k=0 gives the two base angles. Tangent has period , so its general solution adds , not . An interval question ("solve on ") wants the finite list . A general-solution question ("find all real x ") wants the formula above. Same work — you just stop at a different moment. 3. Getting There: Factor, Don't Divide Real equations rarely arrive pre-isolated. Take . It is tempting to divide both sides by — but that destroys every solution where . Instead, move everything to one side and factor . Each factor set to zero is its own little equation; the roots stamp onto the circle. Factor one: gives x = 0 and on .

This is the written version of the interactive lesson above. See the full Calculus Readiness course.