Trigonometry and Calculus Reference

This set of flashcards covers essential Trigonometry identities, differentiation rules, and basic integration formulas as presented in the lecture reference sheets.

Sine (Right Triangle Definition)

$$\sin \theta = \frac{\text{opp}}{\text{hyp}}$$

Cosecant (Right Triangle Definition)

$$\csc \theta = \frac{\text{hyp}}{\text{opp}}$$

Cosine (Right Triangle Definition)

$$\cos \theta = \frac{\text{adj}}{\text{hyp}}$$

Secant (Right Triangle Definition)

$$\sec \theta = \frac{\text{hyp}}{\text{adj}}$$

Tangent (Right Triangle Definition)

$$\tan \theta = \frac{\text{opp}}{\text{adj}}$$

Cotangent (Right Triangle Definition)

$$\cot \theta = \frac{\text{adj}}{\text{opp}}$$

Sine (Circular Function Definition)

$$\sin \theta = \frac{y}{r}$$

Cosine (Circular Function Definition)

$$\cos \theta = \frac{x}{r}$$

Tangent (Circular Function Definition)

$$\tan \theta = \frac{y}{x}$$

Pythagorean Identity (Fundamental)

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

Pythagorean Identity (Tangent/Secant)

$$1 + \tan^2 x = \sec^2 x$$

Pythagorean Identity (Cotangent/Cosecant)

$$1 + \cot^2 x = \csc^2 x$$

Cofunction Identity for Sine

$$\sin\left(\frac{\pi}{2} - x\right) = \cos x$$

Cofunction Identity for Tangent

$$\tan\left(\frac{\pi}{2} - x\right) = \cot x$$

Even/Odd Identity for Cosine

$$\cos(-x) = \cos x$$

Even/Odd Identity for Sine

$$\sin(-x) = -\sin x$$

Sum Formula for Sine

$$\sin(u + v) = \sin u \cos v + \cos u \sin v$$

Difference Formula for Cosine

$$\cos(u - v) = \cos u \cos v + \sin u \sin v$$

Double-Angle Formula for Sine

$$\sin 2u = 2 \sin u \cos u$$

Double-Angle Formulas for Cosine

$$\cos 2u = \cos^2 u - \sin^2 u = 2 \cos^2 u - 1 = 1 - 2 \sin^2 u$$

Power-Reducing Formula for Sine

$$\sin^2 u = \frac{1 - \cos 2u}{2}$$

Sum-to-Product Formula (sin u + sin v)

$$\sin u + \sin v = 2 \sin\left(\frac{u+v}{2}\right) \cos\left(\frac{u-v}{2}\right)$$

Derivative of a Constant

$$\frac{d}{dx}[c] = 0$$

Product Rule for Differentiation

$$\frac{d}{dx}[uv] = uv' + vu'$$

Quotient Rule for Differentiation

$$\frac{d}{dx}\left[\frac{u}{v}\right] = \frac{vu' - uv'}{v^2}$$

Chain Rule (Power Function)

$$\frac{d}{dx}[u^n] = nu^{n-1}u'$$

Derivative of Natural Logarithm

$$\frac{d}{dx}[\ln u] = \frac{u'}{u}$$

Derivative of Exponential Functions (Base e)

$$\frac{d}{dx}[e^u] = e^u u'$$

Derivative of Sine

$$\frac{d}{dx}[\sin u] = (\cos u)u'$$

Derivative of Cosine

$$\frac{d}{dx}[\cos u] = -(\sin u)u'$$

Derivative of Tangent

$$\frac{d}{dx}[\tan u] = (\sec^2 u)u'$$

Derivative of arctan u

$$\frac{d}{dx}[\arctan u] = \frac{u'}{1 + u^2}$$

Power Rule for Integration

$$\int u^n du = \frac{u^{n+1}}{n+1} + C, n \neq -1$$

Integral of 1/u

$$\int \frac{du}{u} = \ln|u| + C$$

Integral of Cosine

$$\int \cos u du = \sin u + C$$

Integral of Sine

$$\int \sin u du = -\cos u + C$$

Integral of sec^2 u

$$\int \sec^2 u du = \tan u + C$$

Integration Formula for arcsin

$$\int \frac{du}{\sqrt{a^2 - u^2}} = \arcsin \frac{u}{a} + C$$