Overview
This lecture covers the definitions, properties, and domain restrictions of the six inverse trigonometric functions, focusing on arcsin, arccos, and arctan, including their evaluation and graphical representations.
Invertibility & Domain Restrictions
- Trigonometric functions are not one-to-one, so they require domain restrictions to become invertible.
- Sine is restricted to domain ([-π/2, π/2]) to define its inverse.
- Cosine is restricted to domain ([0, π]) to define its inverse.
- Tangent is restricted to domain ((-π/2, π/2)) (not including endpoints) to define its inverse.
- Restricting domains ensures all potential function outputs are included in the range.
Inverse Sine (Arcsin)
- Defined as (y = \arcsin(x)) if and only if (x = \sin(y)).
- Domain: ([-1, 1]). Range: ([-π/2, π/2]).
- The output of arcsin is an angle (in radians).
- To compute, ask “What angle has a sine value of (x)?”
- Example evaluations: (\arcsin(1) = π/2, \arcsin(-1/2) = -π/6, \arcsin(\sqrt{3}/2) = π/3).
- Cancellation: (\arcsin(\sin(θ)) = θ) only if (\theta ) is in ([-π/2, π/2]).
Inverse Cosine (Arccos)
- Defined as (y = \arccos(x)) if and only if (x = \cos(y)).
- Domain: ([-1, 1]). Range: ([0, π]).
- The output of arccos is an angle (in radians).
- To compute, ask “What angle has a cosine value of (x)?”
- Example evaluations: (\arccos(0) = π/2, \arccos(-\sqrt{2}/2) = 3π/4).
- Cancellation: (\arccos(\cos(θ)) = θ) only if (θ) is in ([0, π]).
Inverse Tangent (Arctan)
- Defined as (y = \arctan(x)) if and only if (x = \tan(y)).
- Domain: ((-\infty, \infty)). Range: ((-π/2, π/2)) (not including endpoints).
- Use the phrase: “Arctan of (x) is the angle whose tangent is (x).”
- Example evaluations: (\arctan(1) = π/4, \arctan(-\sqrt{3}) = -π/3).
- Cancellation: (\arctan(\tan(θ)) = θ) only if θ is in ((-π/2, π/2)).
Other Inverse Trig Functions
- Arccsc: (\text{Domain: } x \leq -1 \text{ or } x \geq 1; \text{ Range: } [-π/2, π/2], y \neq 0).
- Arcsec: (\text{Domain: } x \leq -1 \text{ or } x \geq 1; \text{ Range: } [0, π], y \neq π/2).
- Arccot: (\text{Domain: } (-\infty, \infty); \text{ Range: } (0, π)).
- Evaluate by asking for the angle whose corresponding reciprocal trigonometric value is (x).
Key Terms & Definitions
- Inverse Function — A function that reverses the effect of the original function.
- Arcsin/Arccos/Arctan — The angles whose sine, cosine, or tangent is a given value.
- Domain — Set of allowable input values.
- Range — Set of possible output values.
- Cancellation Equations — Formulas showing function-inverse pairs return the original value (within restricted domains).
Action Items / Next Steps
- Complete homework on inverse trigonometric functions.
- Refer to the textbook for additional examples and explanations.
- Prepare for Section 7.1 before returning to Section 6.2.