The Range of arccos
Part of the Trig Equations and Inverses section of Coddy's Trigonometry journey. Lesson 7 of 56.
Arccos cannot use arcsin's range: cosine is the same at 60° and −60°, so it would meet each value twice there. Arccos answers from [0°, 180°] instead, the top half of the circle, where cosine falls from 1 to −1 exactly once.
So arccos is never negative, and a negative input lands in quadrant II. arccos(−1/2) has reference angle 60°, and the point on the left of the top half is 180° − 60°, so arccos(−1/2) = 120°. In radians the range is [0, π].
Challenge
arccos(−√3/2) is the one angle in [0°, 180°] whose cosine is −√3/2. Tap it on the circle, then press Check.
Test yourselfThis lesson includes a short quiz. Start the lesson to answer it and track your progress.
Test yourselfThis lesson includes a short quiz. Start the lesson to answer it and track your progress.
All lessons in Trig Equations and Inverses
1Your First Trig Equations
One Equation, Two AnglesReading Signs on the CircleThe Reference AngleRecap: First Equations4Compositions
Draw the TriangleThe Outer FunctionA Negative InsideWhen Undoing Does Not UndoRecap: Compositions2Inverses and Their Ranges
Why arcsin Gives One AngleNegative Inputs to arcsinThe Range of arccosThe Range of arctanRecap: Inverse Ranges5Equations on a Domain
A Negative CosineTangent EquationsEquations in RadiansTwo Turns, Four AnswersRecap: On a Domain