Analytic Trigonometry Concepts and Identities

Vocabulary and identity flashcards covering analytic trigonometry, including sum/difference, double-angle, triple-angle, and linear combinations of sines and cosines.

Sine Sum Formula

The trigonometric identity used to expand the sine of two added angles, defined as $\sin(\theta + \phi) = \sin(\theta)\cos(\phi) + \cos(\theta)\sin(\phi)$.

Sum of Sines and Cosines Form

The linear combination form $a\sin(x) + b\cos(x)$, which can be expressed as a single sine function $k\sin(x + \phi)$.

Amplitude Factor $k$

In the linear combination $a\sin(x) + b\cos(x) = k\sin(x + \phi)$, the constant factor defined by $k = \sqrt{a^2 + b^2}$.

Phase Angle $\phi$ Equations

In rewriting $a\sin(x) + b\cos(x)$ as $k\sin(x + \phi)$, the angle $\phi$ that must satisfy both $\cos(\phi) = \frac{a}{\sqrt{a^2 + b^2}}$ and $\sin(\phi) = \frac{b}{\sqrt{a^2 + b^2}}$.

Double-Angle Formula for Sine

The trigonometric identity that expresses sine of twice an angle, given by $\sin(2x) = 2\sin(x)\cos(x)$.

Double-Angle Formulas for Cosine

The trigonometric identities for cosine of twice an angle, written as $\cos(2x) = \cos^2(x) - \sin^2(x)$, $1 - 2\sin^2(x)$, or $2\cos^2(x) - 1$.

Triple-Angle Formula for Cosine

An identity expressing $\cos(3x)$ purely in terms of cosine powers, given by $\cos(3x) = 4\cos^3(x) - 3\cos(x)$.

Pythagorean Identity

The fundamental trigonometric relationship between sine and cosine given by $\cos^2(x) + \sin^2(x) = 1$.

Quadrant II Trigonometric Signs

In quadrant II of the unit circle, $\cos(\theta)$ is negative and $\sin(\theta)$ is positive.

Quadrant III Trigonometric Signs

In quadrant III of the unit circle, both $\cos(\phi)$ and $\sin(\phi)$ are negative.