Cambridge O Level Additional Mathematics · Syllabus 4037 · Trigonometry
Complete Solution Set
What is Complete Solution Set?
A complete solution set is every angle in the stated domain that satisfies the equation, with no angle omitted and none included that does not satisfy it or that lies outside the domain. Because trigonometric functions are periodic and many-to-one, a single calculator value is only the reference angle; the complete set is built by combining that reference angle with the quadrants in which the function has the required sign, and then adding or subtracting whole periods until the domain is exhausted. Completeness is the whole requirement, so a correct method that stops at one angle has not answered the question.
This definition is part of the Trigonometry chapter in Cambridge O Level Additional Mathematics.
Complete Solution Set in context
A calculator's inverse trigonometric value is only a reference angle, never the full answer, because sine, cosine and tangent are many-to-one: each attainable value is repeated in two quadrants per revolution, and again every period after that. Solving an equation such as \(\sin\theta=0.5\) over \(0^\circ\le\theta\le360^\circ\) means combining that one reference angle, \(30^\circ\), with the quadrants in which sine is positive to get \(\theta=30^\circ\) and \(\theta=150^\circ\); tangent instead repeats every \(180^\circ\), not \(360^\circ\). A complete solution set has no angle missing and none outside the stated domain, so stopping after the first quadrant answers only half the question.
Questions students ask about Complete Solution Set
Why isn't the value your calculator gives for \(\sin^{-1}(0.5)\) the full answer?
Because \(\sin^{-1}\) is defined to return exactly one value, the reference angle, while sine is many-to-one and takes the value \(0.5\) at angles in two quadrants per revolution. \(\sin^{-1}(0.5)=30^\circ\) is the seed, not the answer; the complete solution set over \(0^\circ\) to \(360^\circ\) also includes \(150^\circ\), found from the quadrant where sine is positive. Combine the reference angle with quadrant signs and periodicity before you stop.