Overview
This lecture explains how to evaluate inverse trigonometric functions (arcsin, arccos, arctan), details their ranges and domains, and provides step-by-step examples for common input values.
Evaluating Inverse Sine (arcsin)
- arcsin(x) gives the angle whose sine is x, with a range from -π/2 to π/2 (−90° to 90°), existing in quadrants 1 and 4.
- arcsin(1/2) = π/6 (30°), not 5π/6, because π/6 is within the range.
- arcsin(√3/2) = π/3 (60°), since 60° is in quadrant 1.
- arcsin(−1/2) = −π/6 (−30°), as −30° is in quadrant 4 and within the range.
- arcsin(−√2/2) = −π/4 (−45°), quadrant 4 and within the range.
- arcsin(0) = 0.
- arcsin(1) = π/2 (90°).
- arcsin(−1) = −π/2 (−90°).
Evaluating Inverse Cosine (arccos)
- arccos(x) produces the angle whose cosine is x, with a range from 0 to π (0° to 180°) and exists in quadrants 1 and 2.
- arccos(1/2) = π/3 (60°), as 60° is in the range.
- arccos(−√3/2) = 5π/6 (150°), quadrant 2 and within the range.
- arccos(−√2/2) = 3π/4 (135°), quadrant 2.
- arccos(0) = π/2 (90°).
- arccos(1) = 0.
- arccos(−1) = π.
Evaluating Inverse Tangent (arctan)
- arctan(x) finds the angle whose tangent is x, with a range from -π/2 to π/2 (−90° to 90°) in quadrants 1 and 4.
- arctan(0) = 0.
- arctan(1) = π/4 (45°).
- arctan(−1) = −π/4 (−45°).
- arctan(√3) = π/3 (60°), quadrant 1.
- arctan(−√3/3) = −π/6 (−30°), quadrant 4.
Summary of Ranges and Quadrants
- arcsin/arctan: Range is from –π/2 to π/2; only quadrants 1 and 4.
- arccos: Range is 0 to π; only quadrants 1 and 2.
- Do not select answers in quadrants 2 or 3 for arcsin/arctan, or in quadrants 3 or 4 for arccos.
Compositions with Inverse Trig Functions
- arcsin(sin(θ)) = θ when θ is in [–π/2, π/2]; if outside, result is the equivalent angle within the range.
Key Terms & Definitions
- Inverse Trigonometric Function — Function giving the angle for a given trigonometric value.
- Range — Set of possible output angles for an inverse function.
- Quadrant — One of four sections of the coordinate plane, affecting trigonometric sign.
Action Items / Next Steps
- Practice evaluating inverse trig functions for common values.
- Memorize the ranges and appropriate quadrants for arcsin, arccos, and arctan.
- Review unit circle values for sine, cosine, and tangent.