Evaluating arcsin(1/2)

Trigonometry · Axiom Academy

EXAMPLE Evaluating arcsin(1/2) Find the angle in [-π/2, π/2] whose sine equals 1/2 Excellent work! You've successfully evaluated arcsin(1/2). Here's what we learned: Understanding arcsin: arcsin(x) asks "what angle θ has sin(θ) = x?" It's the inverse of the sine function. Range restriction is crucial: arcsin always returns angles in [-π/2, π/2]. This ensures each input has exactly one output. Special angles matter: Memorizing that sin(π/6) = 1/2 (or sin(30°) = 1/2) makes these problems quick to solve. Always verify the range: Even if you know sin(5π/6) = 1/2, that angle is NOT in [-π/2, π/2], so it's not the answer to arcsin(1/2). Sign determines quadrant: Since 1/2 > 0, we need a first quadrant angle. If we had arcsin(-1/2), we'd look in the fourth quadrant. This systematic approach works for all inverse trig functions: understand the question, recall the range, identify the special angle, and verify it's in the correct range. Practice with arccos and arctan next!

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