MA 113 - Exam 2 Memorization List

Vocabulary flashcards covering trigonometric identities, properties of natural logarithms, and derivative formulas required for MA 113 Exam 2.

Quotient Identity for Tangent

$$\tan(x) = \frac{\sin(x)}{\cos(x)}$$

Quotient Identity for Cotangent

$$\cot(x) = \frac{\cos(x)}{\sin(x)}$$

Reciprocal Identity for Secant

$$\sec(x) = \frac{1}{\cos(x)}$$

Reciprocal Identity for Cosecant

$$\csc(x) = \frac{1}{\sin(x)}$$

Inverse Trigonometric Functions

Functions including $\arcsin(x)$, $\arccos(x)$, $\arctan(x)$, and $\text{arcsec}(x)$ defined as the inverses of the trigonometric functions with restricted domains.

Key Coordinate Checkpoints on the Unit Circle

Key coordinate points $(\cos(\theta), \sin(\theta))$ corresponding to angles: $\theta = 0 \implies (1, 0)$, $\theta = \frac{\pi}{2} \implies (0, 1)$, $\theta = \pi \implies (-1, 0)$, and $\theta = \frac{3\pi}{2} \implies (0, -1)$.

Primary Pythagorean Identity

$$\sin^2(x) + \cos^2(x) = 1$$

Pythagorean Identity for Tangent and Secant

$1 + \tan^2(x) = \sec^2(x)$ (derived by dividing the primary Pythagorean identity by $\cos^2(x)$).

Pythagorean Identity for Cotangent and Cosecant

$1 + \cot^2(x) = \csc^2(x)$ (derived by dividing the primary Pythagorean identity by $\sin^2(x)$).

Double-Angle Identity for Sine

$$\sin(2x) = 2\sin(x)\cos(x)$$

Double-Angle Identity for Cosine (Three Forms)

$$\cos(2x) = \cos^2(x) - \sin^2(x) = 2\cos^2(x) - 1 = 1 - 2\sin^2(x)$$

Product Rule of Natural Logarithms

$\ln(xy) = \ln(x) + \ln(y)$ (for $x > 0$ and $y > 0$).

Quotient Rule of Natural Logarithms

$\ln\left(\frac{x}{y}\right) = \ln(x) - \ln(y)$ (for $x > 0$ and $y > 0$).

Power Rule of Natural Logarithms

$\ln(x^p) = p\ln(x)$ (for $x > 0$, where $p$ is any real number).

Power Rule for Differentiation

$$\frac{d}{dx}\left[ x^n \right] = n x^{n-1}$$

Derivative of the Exponential Function

$$\frac{d}{dx}\left[ e^x \right] = e^x$$

Derivative of the Natural Logarithm

$$\frac{d}{dx}\left[ \ln(|x|) \right] = \frac{1}{x}$$

Derivative of Sine

$$\frac{d}{dx}\left[ \sin(x) \right] = \cos(x)$$

Derivative of Cosine

$$\frac{d}{dx}\left[ \cos(x) \right] = -\sin(x)$$

Derivative of Tangent

$$\frac{d}{dx}\left[ \tan(x) \right] = \sec^2(x)$$

Derivative of Cotangent

$$\frac{d}{dx}\left[ \cot(x) \right] = -\csc^2(x)$$

Derivative of Secant

$$\frac{d}{dx}\left[ \sec(x) \right] = \sec(x)\tan(x)$$

Derivative of Cosecant

$$\frac{d}{dx}\left[ \csc(x) \right] = -\csc(x)\cot(x)$$

Derivative of Arcsine

$$\frac{d}{dx}\left[ \arcsin(x) \right] = \frac{1}{\sqrt{1 - x^2}}$$

Derivative of Arccosine

$$\frac{d}{dx}\left[ \arccos(x) \right] = -\frac{1}{\sqrt{1 - x^2}}$$

Derivative of Arctangent

$$\frac{d}{dx}\left[ \arctan(x) \right] = \frac{1}{1 + x^2}$$

Derivative of Arcsecant

$$\frac{d}{dx}\left[ \text{arcsec}(x) \right] = \frac{1}{x\sqrt{x^2 - 1}}$$