Cambridge O Level Additional Mathematics · Syllabus 4037 · Trigonometry
Complete Angle
What is Complete Angle?
A complete angle is the entire expression that a trigonometric function acts upon, such as two theta plus thirty degrees in the equation sine of two theta plus thirty degrees equals a half. It must be solved for as a single quantity. Because the complete angle changes faster than the variable itself, its domain is wider than the stated domain: substituting the endpoints of the given range into the expression gives the expanded domain in which every solution must be sought, and only after all of those are found is each one converted back to the variable. Solving for the variable first and then trying to add periods is the standard route to a missing solution.
This definition is part of the Trigonometry chapter in Cambridge O Level Additional Mathematics.
Questions students ask about Complete Angle
How do you solve an equation with a complete angle such as \(2\theta+30^\circ\)?
Expand the domain first: substitute the endpoints of the stated range for \(\theta\) into the complete angle to get the wider domain the complete angle itself must range over, then solve for the complete angle inside that expanded domain. Only after every solution for the complete angle has been found do you subtract \(30^\circ\) and divide by \(2\) to return to \(\theta\). Solving for \(\theta\) first and adding periods afterwards is the standard route to a missing solution.