Comprehensive Trigonometry Formula Guide

Fundamental Compound Angle Formulas

The fundamental addition and subtraction formulas for trigonometric functions allow for the expansion of sine, cosine, tangent, and cotangent when the argument is a sum or difference of two angles. For sine, the expansions are given as: $\sin(A+B) = \sin(A)\cos(B) + \cos(A)\sin(B)$$\sin(A-B) = \sin(A)\cos(B) - \cos(A)\sin(B)$

For cosine, the signs are reversed between the argument and the expansion: $\cos(A+B) = \cos(A)\cos(B) - \sin(A)\sin(B)$$\cos(A-B) = \cos(A)\cos(B) + \sin(A)\sin(B)$

The tangent expansions involve ratios of the tangent of individual angles: $\tan(A+B) = \frac{\tan(A) + \tan(B)}{1 - \tan(A)\tan(B)}$$\tan(A-B) = \frac{\tan(A) - \tan(B)}{1 + \tan(A)\tan(B)}$

The cotangent expansions are structured as follows: $\cot(A+B) = \frac{\cot(A)\cot(B) - 1}{\cot(B) + \cot(A)}$$\cot(A-B) = \frac{\cot(A)\cot(B) + 1}{\cot(B) - \cot(A)}$

Product-to-Sum and Sum-to-Product Transformations

Product-to-sum formulas are used to transform a product of trigonometric functions into a sum or difference, which is often easier to integrate or differentiate. The identities provided are: $2\sin(A)\cos(B) = \sin(A+B) + \sin(A-B)$$2\cos(A)\cos(B) = \cos(A+B) + \cos(A-B)$$2\sin(B)\cos(A) = \sin(A+B) - \sin(A-B)$$2\sin(A)\sin(B) = \cos(A-B) - \cos(A+B)$

Sum-to-product formulas perform the inverse transformation, converting the sum or difference of two sine or cosine functions into a product: $\sin(C) + \sin(D) = 2\sin\left(\frac{C+D}{2}\right)\cos\left(\frac{C-D}{2}\right)$$\sin(C) - \sin(D) = 2\cos\left(\frac{C+D}{2}\right)\sin\left(\frac{C-D}{2}\right)$$\cos(C) + \cos(D) = 2\cos\left(\frac{C+D}{2}\right)\cos\left(\frac{C-D}{2}\right)$$\cos(C) - \cos(D) = -2\sin\left(\frac{C+D}{2}\right)\sin\left(\frac{C-D}{2}\right)$

Advanced Power and Multiple Angle Identities

The transcript identifies specialized identities for higher powers of sine and cosine: $\sin^4(\theta) + \cos^4(\theta) = 1 - 2\sin^2(\theta)\cos^2(\theta)$$\sin^6(\theta) + \cos^6(\theta) = 1 - 3\sin^2(\theta)\cos^2(\theta)$

Double angle formulas represent functions of $2A$ in terms of functions of $A$: $\sin(2A) = 2\sin(A)\cos(A) = \frac{2\tan(A)}{1 + \tan^2(A)}$$\cos(2A) = \cos^2(A) - \sin^2(A) = 1 - 2\sin^2(A) = 2\cos^2(A) - 1 = \frac{1 - \tan^2(A)}{1 + \tan^2(A)}$

From these, power reduction identities for squared terms can be derived: $\sin^2(A) = \frac{1 - \cos(2A)}{2}$$\cos^2(A) = \frac{1 + \cos(2A)}{2}$

Tangent and cotangent double angle formulas include: $\tan(2A) = \frac{2\tan(A)}{1 - \tan^2(A)}$$\cot(2A) = \frac{\cot^2(A) - 1}{2\cot(A)}$

Specific relationships between tangent and cotangent are highlighted: $\cot(\theta) - \tan(\theta) = 2\cot(2\theta)$$\cot(\theta) + \tan(\theta) = 2\csc(2\theta)$$\cot(\theta) = \cot(2\theta) + \csc(2\theta)$

Triple angle formulas include: $\sin(3\theta) = 3\sin(\theta) - 4\sin^3(\theta)$$\cos(3\theta) = 4\cos^3(\theta) - 3\cos(\theta)$$\tan(3\theta) = \frac{3\tan(\theta) - \tan^3(\theta)}{1 - 3\tan^2(\theta)}$

Specialized Product Formulas and Specific Angles

The following products involving angles shifted by $60^\circ$ provide compact results: $\sin(60^\circ - \theta)\sin(\theta)\sin(60^\circ + \theta) = \frac{1}{4}\sin(3\theta)$$\cos(60^\circ - \theta)\cos(\theta)\cos(60^\circ + \theta) = \frac{1}{4}\cos(3\theta)$$\tan(60^\circ - \theta)\tan(\theta)\tan(60^\circ + \theta) = \tan(3\theta)$$\cot(60^\circ - \theta)\cot(\theta)\cot(60^\circ + \theta) = \cot(3\theta)$

Additional identities for products of sums and differences are: $\sin(A+B)\sin(A-B) = \sin^2(A) - \sin^2(B)$$\cos(A+B)\cos(A-B) = \cos^2(A) - \sin^2(B) = \cos^2(B) - \sin^2(A)$

Quadrant Transformations, Periodicity, and Half Angles

Pythagorean identities represent the fundamental relationships on the unit circle: $\sin^2(\theta) + \cos^2(\theta) = 1$$\sec^2(\theta) - \tan^2(\theta) = 1$$\csc^2(\theta) - \cot^2(\theta) = 1$

Behavior of functions with negative arguments (even/odd properties): $\sin(-\theta) = -\sin(\theta)$$\cos(-\theta) = \cos(\theta)$$\tan(-\theta) = -\tan(\theta)$$\cot(-\theta) = -\cot(\theta)$$\sec(-\theta) = \sec(\theta)$$\csc(-\theta) = -\csc(\theta)$

Transformations based on multiples of $\frac{\pi}{2}$ (Cofunctions): $\sin\left(\frac{\pi}{2} - \theta\right) = \cos(\theta)$$\cos\left(\frac{\pi}{2} - \theta\right) = \sin(\theta)$$\tan\left(\frac{\pi}{2} - \theta\right) = \cot(\theta)$$\sec\left(\frac{\pi}{2} - \theta\right) = \csc(\theta)$$\csc\left(\frac{\pi}{2} - \theta\right) = \sec(\theta)$

For general angles involving $\frac{(2n+1)\pi}{2} \pm \theta$, the function changes to its cofunction (sine becomes cosine, etc.). For transformations involving $n\pi \pm \theta$, the function remains the same (sine stays sine), with the sign determined by the quadrant.

Half-angle formulas describe functions of $\frac{\theta}{2}$ in terms of $\theta$: $\sin(\theta) = 2\sin\left(\frac{\theta}{2}\right)\cos\left(\frac{\theta}{2}\right)$$\cos(\theta) = \cos^2\left(\frac{\theta}{2}\right) - \sin^2\left(\frac{\theta}{2}\right) = 2\cos^2\left(\frac{\theta}{2}\right) - 1 = 1 - 2\sin^2\left(\frac{\theta}{2}\right)$$\tan(\theta) = \frac{2\tan\left(\frac{\theta}{2}\right)}{1 - \tan^2\left(\frac{\theta}{2}\right)}$

Trigonometric Series Summation

Summation of sine and cosine series where the angles are in arithmetic progression: $\sin(\alpha) + \sin(\alpha + \beta) + \sin(\alpha + 2\beta) + \dots + \sin(\alpha + (n-1)\beta) = \frac{\sin\left(\frac{n\beta}{2}\right)}{\sin\left(\frac{\beta}{2}\right)} \times \sin\left(\alpha + \frac{(n-1)\beta}{2}\right)$$\cos(\alpha) + \cos(\alpha + \beta) + \cos(\alpha + 2\beta) + \dots + \cos(\alpha + (n-1)\beta) = \frac{\sin\left(\frac{n\beta}{2}\right)}{\sin\left(\frac{\beta}{2}\right)} \times \cos\left(\alpha + \frac{(n-1)\beta}{2}\right)$

For a specific series of sine terms with increasing frequency: $\sin(\theta) + \sin(2\theta) + \sin(3\theta) + \dots + \sin(N\theta) = \frac{\sin\left(\frac{(N+1)\theta}{2}\right)\sin\left(\frac{N\theta}{2}\right)}{\sin\left(\frac{\theta}{2}\right)}$

Product of cosine terms with doubling angles: $\cos(\theta)\cos(2\theta)\cos(2^2\theta) \dots \cos(2^{n-1}\theta) = \frac{\sin(2^n\theta)}{2^n\sin(\theta)}$

Conditional Trigonometry (A + B + C = \pi)

In the context of a triangle where $A+B+C = \pi$, the following identities hold: $\sin(2A) + \sin(2B) + \sin(2C) = 4\sin(A)\sin(B)\sin(C)$$\cos(2A) + \cos(2B) + \cos(2C) = -1 - 4\cos(A)\cos(B)\cos(C)$$\tan(A) + \tan(B) + \tan(C) = \tan(A)\tan(B)\tan(C)$$\cot(A)\cot(B) + \cot(B)\cot(C) + \cot(A)\cot(C) = 1$$\sin(A) + \sin(B) + \sin(C) = 4\cos\left(\frac{A}{2}\right)\cos\left(\frac{B}{2}\right)\cos\left(\frac{C}{2}\right)$

Additional conditional constraints include: If $A+B = \pi$, then $\cos(A) + \cos(B) = 0$. If $A+B = \frac{\pi}{2}$, then $\tan(A)\tan(B) = 1$ and $\cot(A)\cot(B) = 1$.

Trigonometric Equations and General Solutions

Principal solutions are found within the range $[0, 2\pi)$. General solutions for any integer $n \in I$ are as follows:

If $\sin(\theta) = 0$, then $\theta = n\pi$ If $\cos(\theta) = 0$, then $\theta = (2n+1)\frac{\pi}{2}$ If $\tan(\theta) = 0$, then $\theta = n\pi$ If $\cot(\theta) = 0$, then $\theta = (2n+1)\frac{\pi}{2}$

For equations where the function equals a non-zero value $\alpha$: $\sin(\theta) = \sin(\alpha) \Rightarrow \theta = n\pi + (-1)^n\alpha$$\cos(\theta) = \cos(\alpha) \Rightarrow \theta = 2n\pi \pm \alpha$$\tan(\theta) = \tan(\alpha) \Rightarrow \theta = n\pi + \alpha$$\cot(\theta) = \cot(\alpha) \Rightarrow \theta = n\pi + \alpha$$\sec(\theta) = \sec(\alpha) \Rightarrow \theta = 2n\pi \pm \alpha$$\csc(\theta) = \csc(\alpha) \Rightarrow \theta = n\pi + (-1)^n\alpha$

For squared trigonometric equations: $\sin^2(\theta) = \sin^2(\alpha), \cos^2(\theta) = \cos^2(\alpha), \text{ or } \tan^2(\theta) = \tan^2(\alpha) \Rightarrow \theta = n\pi \pm \alpha$