Trig Graphing and Inverse Functions Notes (5.3–5.5)

Amplitude of a Transformed Sine/Cosine Graph

The absolute value of the coefficient 'a' in the function y = a\sin(bx - c). It represents half the distance between the maximum and minimum values of the function: A = \|a\|.

Period of a Transformed Sine/Cosine Graph

The length of one complete cycle of the graph, calculated as T = \frac{2\pi}{\|b\|} for functions of the form y = a\sin(bx - c).

Phase Shift (Horizontal Shift)

The horizontal translation of a trigonometric graph. For y = a\sin(bx - c), the shift is \Delta x = \frac{c}{\|b\|} to the right.

General Graphing Plan for Trig Functions

1. Apply vertical stretch/reflection (amplitude). 2. Adjust period (horizontal compression/stretch). 3. Apply phase shift (horizontal translation).

Common Pitfalls in Graphing Trig Functions

Forgetting reflection for negative amplitude, miscomputing phase shift (not dividing by 'b'), mixing up left/right direction of phase shift.

Tangent Graph Asymptotes

Vertical asymptotes occur at all odd multiples of \frac{\pi}{2} (i.e., x = \frac{\pi}{2} + k\pi where k is an integer).

Cotangent Graph Asymptotes

Vertical asymptotes occur at all integer multiples of \pi (i.e., x = k\pi where k is an integer).

Secant Graph Asymptotes

Vertical asymptotes occur where \cos x = 0 (i.e., at x = \frac{\pi}{2} + k\pi).

Cosecant Graph Asymptotes

Vertical asymptotes occur where \sin x = 0 (i.e., at x = k\pi).

Inverse Trigonometric Functions - Need for Restricted Domain

To ensure the inverse is a function (passes the horizontal line test), the original trigonometric function must be restricted to a suitable one-to-one domain.

Domain and Range for Sine Inverse (\arcsin(x) \text{ or } \sin^{-1}(x))

Domain: x \in [-1, 1]. Range: y \in [-\tfrac{\pi}{2}, \tfrac{\pi}{2}].

Domain and Range for Cosine Inverse (\arccos(x) \text{ or } \cos^{-1}(x))

Domain: x \in [-1, 1]. Range: y \in [0, \pi].

Domain and Range for Tangent Inverse (\arctan(x) \text{ or } \tan^{-1}(x))

Domain: x \in \mathbb{R}. Range: y \in (-\tfrac{\pi}{2}, \tfrac{\pi}{2}).

Inverse Cancellation: Inverse Inside (e.g., \sin(\arcsin(x)))

If the inverse function is inside the argument, they cancel straightforwardly, e.g., \sin(\arcsin(x)) = x, provided x is in the domain of the inner inverse function.

Inverse Cancellation: Inverse Outside (e.g., \arcsin(\sin(x)))

If the inverse function is outside, you cannot simply cancel to x. You must map x back into the principal range of the outer inverse function (e.g., for \arcsin(\sin(x)), the result must be in [-\tfrac{\pi}{2}, \tfrac{\pi}{2}]).

Arc Notation

A common alternative notation for inverse trigonometric functions (e.g., arc sine for \sin^{-1}(x), arc cosine for \cos^{-1}(x), etc.).

Handling Inverse Reciprocal Functions (e.g., \text{arcsec(x)})

Convert to corresponding inverse sine/cosine using reciprocal identities, e.g., \text{arcsec}(x) = \arccos(\frac{1}{x}), respecting domain/range.

Principal Range for Sine Inverse (Quadrant Heuristic)

The result of \arcsin(x) must be an angle in Quadrant I or IV (i.e., [-\tfrac{\pi}{2}, \tfrac{\pi}{2}]).

Principal Range for Cosine Inverse (Quadrant Heuristic)

The result of \arccos(x) must be an angle in Quadrant I or II (i.e., [0, \pi]).

Principal Range for Tangent Inverse (Quadrant Heuristic)

The result of \arctan(x) must be an angle in Quadrant I or IV (i.e., (-\tfrac{\pi}{2}, \tfrac{\pi}{2})).