Trigonometry Formulas and Constants

A collection of trigonometric identities, addition formulas, factorization methods, and specific angle values extracted from the lecture notes.

Addition Formula for $\\sin(A+B)$

$$\\sin(A+B) = \\sin(A)\\cos(B) + \\cos(A)\\sin(B)$$

Addition Formula for $\\cos(A+B)$

$$\\cos(A+B) = \\cos(A)\\cos(B) - \\sin(A)\\sin(B)$$

Supplementary Angle of $\\cos(\\pi - \\theta)$

$$\\cos(\\pi - \\theta) = -\\cos(\\theta)$$

Complementary Angle of $\\tan(\\frac{\\pi}{2} - \\theta)$

$$\\tan(\\frac{\\pi}{2} - \\theta) = \\cot(\\theta)$$

Tangent Addition Formula ($\\tan(A+B)$)

$$\\tan(A+B) = \\frac{\\tan(A) + \\tan(B)}{1 - \\tan(A)\\tan(B)}$$

Exact value of $\\sin(15^{\\circ})$

$$\\frac{\\sqrt{3}-1}{2\\sqrt{2}}$$

Exact value of $\\cos(15^{\\circ})$

$$\\frac{\\sqrt{3}+1}{2\\sqrt{2}}$$

Defactorization Formula for $2\\sin(A)\\cos(B)$

$$\\sin(A+B) + \\sin(A-B)$$

Defactorization Formula for $2\\cos(A)\\cos(B)$

$$\\cos(A+B) + \\cos(A-B)$$

Factorization Formula for $\\sin(C) - \\sin(D)$

$$2\\cos\\left(\\frac{C+D}{2}\\right)\\sin\\left(\\frac{C-D}{2}\\right)$$

Factorization Formula for $\\cos(C) - \\cos(D)$

$$2\\sin\\left(\\frac{C+D}{2}\\right)\\sin\\left(\\frac{D-C}{2}\\right)$$

Exact value of $\\sin(18^{\\circ})$

$$\\frac{\\sqrt{5}-1}{4}$$

Double Angle Formula for $\\sin(2\\theta)$

$$2\\sin(\\theta)\\cos(\\theta)$$

Double Angle Formula for $\\cos(2\\theta)$ in terms of $\\sin$

$$1 - 2\\sin^2(\\theta)$$

Relation for $1 + \\cos(2\\theta)$

$$2\\cos^2(\\theta)$$

Triple Angle Formula for $\\sin(3\\theta)$

$$3\\sin(\\theta) - 4\\sin^3(\\theta)$$

Triple Angle Formula for $\\cos(3\\theta)$

$$4\\cos^3(\\theta) - 3\\cos(\\theta)$$

Special Product Formula for Sine

$$\\sin(\\theta)\\sin(60^{\\circ}-\\theta)\\sin(60^{\\circ}+\\theta) = \\frac{1}{4}\\sin(3\\theta)$$

Cosine Double Angle Product Series

$$\\cos(\\theta)\\cos(2\\theta)\\cos(4\\theta)... = \\frac{\\sin(2^n\\theta)}{2^n\\sin(\\theta)}$$

Sum of Sines for Angles in AP

$$\\sum_{k=0}^{n-1} \\sin(a+kd) = \\frac{\\sin\\left(\\frac{nd}{2}\\right)}{\\sin\\left(\\frac{d}{2}\\right)} \\sin\\left(a + \\frac{(n-1)d}{2}\\right)$$