Basic Trigonometric Equations

Algebra 2 · Axiom Academy

LESSON Basic Trigonometric Equations Solving sin θ = a, cos θ = a and tan θ = a by reading every solution straight off the unit circle. On the unit circle, is the height — the y -coordinate — of the point at angle . To solve , find the reference angle , then take the two angles in whose height equals a : one in each quadrant where the sign matches. lands in Quadrant I; its partner is the reflection into Quadrant II. Reference angle (Quadrant I). Quadrant II gives . is the horizontal position — the x -coordinate. Solving means finding where the vertical line x = a cuts the circle: two points symmetric about the x -axis , so the second angle is minus the first. lands in Quadrant I; its partner is the reflection into Quadrant IV. Reference angle (Quadrant I). Quadrant IV gives . is the slope of the radius. A single straight line of that slope through the centre pierces the circle at two opposite points — half a turn apart. That is why tangent's period is , not : its two solutions differ by exactly . Every integer n gives another solution, spaced apart. Reference angle . Add the period once: . Because the values repeat, we capture every solution by adding whole multiples of the period to a base solution: for sine and cosine, for tangent. The integer n runs over all of , so the solution set is infinite. Sine and cosine never leave the interval [-1, 1] , so when the horizontal or vertical line misses the circle entirely. Tangent has no such limit — is solvable for every real a .

This is the written version of the interactive lesson above. See the full Algebra 2 course.