Overview
This lecture covers inverse trigonometric functions, focusing on the arc cosine and arc tangent, their domains, graph properties, and geometric interpretations.
Inverse Cosine (Arc Cosine)
- The inverse of cosine is called arc cosine, written as arccos(x) or cos⁻¹(x).
- To have an inverse, cosine's domain is restricted to [0, π], where it passes the horizontal line test.
- Arccos(x) returns the angle in [0, π] whose cosine is x.
- On the graph, key points (0,1), (π/2,0), and (π, -1) are reflected over the line y = x to create the arccos graph.
- Inverse notation (cos⁻¹) does not mean reciprocal; it indicates the inverse function.
- Calculators can find values like arccos(0.7) ≈ 0.795 radians.
- A right triangle representing arccos(0.7) has adjacent/hypotenuse = 0.7; angle = 0.795 radians.
- Arccos(x) is only defined for x in [-1, 1]; inputting values outside this range (e.g., arccos(2)) yields a math error as it is not possible for a triangle.
Inverse Tangent (Arc Tangent)
- The tangent function passes the horizontal line test only on (-π/2, π/2).
- The inverse tangent, written arctan(x) or tan⁻¹(x), is defined on this interval.
- To find arctan(x), reflect the restricted tangent function over y = x.
- Vertical asymptotes of tangent at x = ±π/2 become horizontal asymptotes at y = ±π/2 for arctan.
- Arctan(1) = π/4 radians (0.785 radians), corresponding to a triangle with sides 1:1.
- Arctan(x) has horizontal asymptotes: as x→∞, arctan(x)→π/2; as x→-∞, arctan(x)→-π/2.
Key Terms & Definitions
- Inverse Function — A function that "undoes" another function, returning the original input.
- Arccos (cos⁻¹) — The inverse of cosine, restricted to [0, π].
- Arctan (tan⁻¹) — The inverse of tangent, restricted to (-π/2, π/2).
- Domain — The set of allowable input values for a function.
- Horizontal Line Test — A graph test to determine if a function is invertible.
- Asymptote — A line that a graph approaches but never touches.
Action Items / Next Steps
- Practice using a calculator to find arccos and arctan for various input values within their domains.
- Review and sketch the graphs of arccos and arctan, highlighting domain restrictions.
- Remember to always use radians in calculations for calculus.