M141 Trigonometry, Exponential, and Logarithmic Functions

Practice vocabulary flashcards covering unit circle trigonometry, parity proofs, exponential and logarithmic functions and graphs, Pythagorean identities, and inverse trigonometric domains based on lecture notes.

Unit Circle Definition of Trigonometric Functions

The definition of trigonometric functions of a real number or arc length $t$ using the terminal point coordinates $(x, y)$ on a unit circle centered at the origin: $\sin(t) = y$, $\cos(t) = x$, $\tan(t) = \frac{y}{x}$, $\csc(t) = \frac{1}{y}$, $\sec(t) = \frac{1}{x}$, and $\cot(t) = \frac{x}{y}$.

Trigonometric Function Values at $t = \pi$

The function values evaluated at terminal point $(-1, 0)$: $\sin(\pi) = 0$, $\cos(\pi) = -1$, $\tan(\pi) = 0$, $\sec(\pi) = -1$, with $\csc(\pi)$ and $\cot(\pi)$ being undefined due to division by zero.

Odd Trigonometric Function (Sine Parity)

A property stating that $\sin(-t) = -\sin(t)$, geometrically shown on the unit circle because the terminal point for $-t$ is $(x, -y)$, making the $y$-coordinate the opposite sign of the terminal point for $t$.

Even Trigonometric Function (Cosine Parity)

A property stating that $\cos(-t) = \cos(t)$, geometrically shown on the unit circle because the terminal point for $-t$ is $(x, -y)$, maintaining the exact same $x$-coordinate as the terminal point for $t$.

Division of Unit Circle into 12 Equal Arcs

Partitioning the circumference of the unit circle into equal arc lengths of $\frac{2\pi}{12} = \frac{\pi}{6}$ radians ($30^\circ$), generating terminal coordinate points at successive multiples of $\frac{\pi}{6}$.

Exponential Function $f(x) = 2^x$

A base-2 exponential function characterized by a domain of $(-\infty, \infty)$, a range of $(0, \infty)$, strictly increasing behavior, and a horizontal asymptote along the line $y = 0$.

Reflection of $f(x) = 2^x$ About the $y$-Axis

The transformed function $f(-x) = 2^{-x} = \left(\frac{1}{2}\right)^x$, which has a domain of $(-\infty, \infty)$, a range of $(0, \infty)$, a horizontal asymptote at $y = 0$, and strictly decreasing behavior.

Reflection of $y = 2^x$ About the Line $y = x$

The inverse relation yielding the logarithmic function $y = \log_2(x)$, which has a domain of $(0, \infty)$, a range of $(-\infty, \infty)$, a vertical asymptote at $x = 0$, and strictly increasing behavior.

Equivalence of Exponential and Logarithmic Forms

The interchangeability of powers and logs given by $b^y = x \iff \log_b(x) = y$, as seen in $2^3 = 8 \iff \log_2(8) = 3$ and $2^6 = 64 \iff \log_2(64) = 6$.

Signs of Trigonometric Functions Across Quadrants

The quadrant sign rule where: in Quadrant I, all three ($\sin$, $\cos$, $\tan$) are positive; in Quadrant II, $\sin$ is positive while $\cos$ and $\tan$ are negative; in Quadrant III, $\tan$ is positive while $\sin$ and $\cos$ are negative; and in Quadrant IV, $\cos$ is positive while $\sin$ and $\tan$ are negative.

Fundamental Pythagorean Identity

The core trigonometric identity $\cos^2(t) + \sin^2(t) = 1$, derived from the equation of the unit circle $x^2 + y^2 = 1$.

Derived Pythagorean Identities

The two identities obtained by dividing $\cos^2(t) + \sin^2(t) = 1$ by $\cos^2(t)$ and $\sin^2(t)$, resulting respectively in $1 + \tan^2(t) = \sec^2(t)$ and $\cot^2(t) + 1 = \csc^2(t)$.

Laws of Logarithms

Algebraic properties used to combine or expand logarithms, consisting of the Product Law $\log_b(xy) = \log_b(x) + \log_b(y)$, the Quotient Law $\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)$, and the Power Law $\log_b(x^k) = k\log_b(x)$.

One-to-One Interval for Sine ($\sin(t) = y$)

The restricted domain interval $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ on which the sine function passes the horizontal line test, making it invertible to define $t = \sin^{-1}(y)$.

One-to-One Interval for Cosine ($\cos(t) = x$)

The restricted domain interval $[0, \pi]$ on which the cosine function passes the horizontal line test, making it invertible to define $t = \cos^{-1}(x)$.

Inverse Trigonometric Function Values

Angles or arc lengths $t$ resulting from evaluating inverse trigonometric relations within their principal branch ranges, such as $\sin^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{6}$, $\cos^{-1}(0) = \frac{\pi}{2}$, and $\sin^{-1}(-1) = -\frac{\pi}{2}$.