Negative Inputs to arcsin
Part of the Trig Equations and Inverses section of Coddy's Trigonometry journey. Lesson 6 of 56.
The range [−90°, 90°] reaches below the x axis by turning clockwise, so a negative input gives a negative angle. arcsin(−1/2) has reference angle 30°, and the point must be on the right half, below the axis: arcsin(−1/2) = −30°.
210° and 330° also have sine −1/2, and 330° even ends at the same point as −30°. Neither is the arcsin, because neither is in the range. The rule arcsin(−c) = −arcsin(c) holds for every c from −1 to 1.
Challenge
Work out x = arcsin(−√3/2) in degrees: write the value as a new line, x = …°, with the degree sign, and press Check.
Try it yourself
Solve for x
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