Day 2: Analytical Trigonometry & Functional Equations 2.1 Product-to-Sum and Sum-to-Product Identities .Sum-to-Product: sin ( x ) + sin ( y ) = 2 sin ( x + y 2 ) cos ( x − y 2 ) sin ( x ) − sin ( y ) = 2 cos ( x + y 2 ) sin ( x − y 2 ) cos ( x ) + cos ( y ) = 2 cos ( x + y 2 ) cos ( x − y 2 ) cos ( x ) − cos ( y ) = − 2 sin ( x + y 2 ) sin ( x − y 2 ) Product-to-Sum: 2 sin ( A ) cos ( B ) = sin ( A + B ) + sin ( A − B ) 2 cos ( A ) sin ( B ) = sin ( A + B ) − sin ( A − B ) 2 cos ( A ) cos ( B ) = cos ( A + B ) + cos ( A − B ) 2 sin ( A ) sin ( B ) = cos ( A − B ) − cos ( A + B ) Angle Addition Base: sin ( A + B ) = sin ( A ) cos ( B ) + cos ( A ) sin ( B ) cos ( A + B ) = cos ( A ) cos ( B ) − sin ( A ) sin ( B ) \begin{aligned}\textbf{Sum-to-Product:} & \\\sin(x) + \sin(y) &= 2 \sin\left(\frac{x+y}{2}\right) \cos\left(\frac{x-y}{2}\right) \\\sin(x) - \sin(y) &= 2 \cos\left(\frac{x+y}{2}\right) \sin\left(\frac{x-y}{2}\right) \\\cos(x) + \cos(y) &= 2 \cos\left(\frac{x+y}{2}\right) \cos\left(\frac{x-y}{2}\right) \\\cos(x) - \cos(y) &= -2 \sin\left(\frac{x+y}{2}\right) \sin\left(\frac{x-y}{2}\right) \\\\\textbf{Product-to-Sum:} & \\2 \sin(A)\cos(B) &= \sin(A+B) + \sin(A-B) \\2 \cos(A)\sin(B) &= \sin(A+B) - \sin(A-B) \\2 \cos(A)\cos(B) &= \cos(A+B) + \cos(A-B) \\2 \sin(A)\sin(B) &= \cos(A-B) - \cos(A+B) \\\\\textbf{Angle Addition Base:} & \\\sin(A+B) &= \sin(A)\cos(B) + \cos(A)\sin(B) \\\cos(A+B) &= \cos(A)\cos(B) - \sin(A)\sin(B)\end{aligned} Sum-to-Product: sin ( x ) + sin ( y ) sin ( x ) − sin ( y ) cos ( x ) + cos ( y ) cos ( x ) − cos ( y ) Product-to-Sum: 2 sin ( A ) cos ( B ) 2 cos ( A ) sin ( B ) 2 cos ( A ) cos ( B ) 2 sin ( A ) sin ( B ) Angle Addition Base: sin ( A + B ) cos ( A + B ) = 2 sin ( 2 x + y ) cos ( 2 x − y ) = 2 cos ( 2 x + y ) sin ( 2 x − y ) = 2 cos ( 2 x + y ) cos ( 2 x − y ) = − 2 sin ( 2 x + y ) sin ( 2 x − y ) = sin ( A + B ) + sin ( A − B ) = sin ( A + B ) − sin ( A − B ) = cos ( A + B ) + cos ( A − B ) = cos ( A − B ) − cos ( A + B ) = sin ( A ) cos ( B ) + cos ( A ) sin ( B ) = cos ( A ) cos ( B ) − sin ( A ) sin ( B ) 2.2 Double / Half-Angle Transformations sin ( 2 θ ) = 2 sin ( θ ) cos ( θ ) cos ( 2 θ ) = cos 2 ( θ ) − sin 2 ( θ ) cos ( 2 θ ) = 2 cos 2 ( θ ) − 1 cos ( 2 θ ) = 1 − 2 sin 2 ( θ ) \begin{aligned}
\sin(2\theta) &= 2\sin(\theta)\cos(\theta) \\
\cos(2\theta) &= \cos^2(\theta) - \sin^2(\theta) \\
\cos(2\theta) &= 2\cos^2(\theta) - 1 \\
\cos(2\theta) &= 1 - 2\sin^2(\theta)
\end{aligned} sin ( 2 θ ) cos ( 2 θ ) cos ( 2 θ ) cos ( 2 θ ) = 2 sin ( θ ) cos ( θ ) = cos 2 ( θ ) − sin 2 ( θ ) = 2 cos 2 ( θ ) − 1 = 1 − 2 sin 2 ( θ ) 2.3 Monotonicity and Injectivity Invariants Initial Value: V start = N ( N − 1 ) 2 Split Operation: Δ V = + 1 Merge Operation: Δ V = − 1 Total Operations: Moves = ∣ V end − V start ∣ \begin{aligned}
\textbf{Initial Value:} \quad V_{\text{start}} &= \frac{N(N-1)}{2} \\
\textbf{Split Operation:} \quad \Delta V &= +1 \\
\textbf{Merge Operation:} \quad \Delta V &= -1 \\
\textbf{Total Operations:} \quad \text{Moves} &= |V_{\text{end}} - V_{\text{start}}|
\end{aligned} Initial Value: V start Split Operation: Δ V Merge Operation: Δ V Total Operations: Moves = 2 N ( N − 1 ) = + 1 = − 1 = ∣ V end − V start ∣ 2.4 Classical Inequalities [ THE AM-GM-HM BOUND ] The standard average is ALWAYS bigger than or equal to the multiplying average. Formula: a + b 2 ≥ a b ≥ 2 1 a + 1 b Shortcut: a + b ≥ 2 a b General Form: x 1 + x 2 + ⋯ + x n n ≥ x 1 ⋅ x 2 … x n n Rule: Equality ONLY happens if all variables are completely identical ( a = b ) . [ THE CAUCHY-SCHWARZ POOL ] Multiplying separate squared pools ALWAYS beats grouping and pairing them up. Identity: ( a 2 + b 2 ) ( x 2 + y 2 ) = ( a x + b y ) 2 + ( a y − b x ) 2 Inequality: ( a 2 + b 2 ) ( x 2 + y 2 ) ≥ ( a x + b y ) 2 Rule: The exact value lost when pairing is the leftover term ( a y − b x ) 2 . [ TITU’S FRACTION] Joining separate fractions into one fraction ALWAYS drops the total value. Formula: a 2 x + b 2 y ≥ ( a + b ) 2 x + y Rule: Add the tops together, square it, and divide by the added bottoms. \begin{aligned}&\textbf{[ THE AM-GM-HM BOUND ]} \\&\text{The standard average is ALWAYS bigger than or equal to the multiplying average.} \\\\&\text{Formula:} \quad \frac{a+b}{2} \geq \sqrt{ab} \geq \frac{2}{\frac{1}{a} + \frac{1}{b}} \\&\text{Shortcut:} \quad a+b \geq 2\sqrt{ab} \\&\text{General Form:} \quad \frac{x_1 + x_2 + \dots + x_n}{n} \geq \sqrt[n]{x_1 \cdot x_2 \dots x_n} \\&\text{Rule:} \quad \text{Equality ONLY happens if all variables are completely identical } (a = b). \\\\\\&\textbf{[ THE CAUCHY-SCHWARZ POOL ]} \\&\text{Multiplying separate squared pools ALWAYS beats grouping and pairing them up.} \\\\&\text{Identity:} \quad (a^2 + b^2)(x^2 + y^2) = (ax + by)^2 + (ay - bx)^2 \\&\text{Inequality:} \quad (a^2 + b^2)(x^2 + y^2) \geq (ax + by)^2 \\&\text{Rule:} \quad \text{The exact value lost when pairing is the leftover term } (ay - bx)^2. \\\\\\&\textbf{[ TITU'S FRACTION]} \\&\text{Joining separate fractions into one fraction ALWAYS drops the total value.} \\\\&\text{Formula:} \quad \frac{a^2}{x} + \frac{b^2}{y} \geq \frac{(a+b)^2}{x+y} \\&\text{Rule:} \quad \text{Add the tops together, square it, and divide by the added bottoms.}\end{aligned} [ THE AM-GM-HM BOUND ] The standard average is ALWAYS bigger than or equal to the multiplying average. Formula: 2 a + b ≥ ab ≥ a 1 + b 1 2 Shortcut: a + b ≥ 2 ab General Form: n x 1 + x 2 + ⋯ + x n ≥ n x 1 ⋅ x 2 … x n Rule: Equality ONLY happens if all variables are completely identical ( a = b ) . [ THE CAUCHY-SCHWARZ POOL ] Multiplying separate squared pools ALWAYS beats grouping and pairing them up. Identity: ( a 2 + b 2 ) ( x 2 + y 2 ) = ( a x + b y ) 2 + ( a y − b x ) 2 Inequality: ( a 2 + b 2 ) ( x 2 + y 2 ) ≥ ( a x + b y ) 2 Rule: The exact value lost when pairing is the leftover term ( a y − b x ) 2 . [ TITU’S FRACTION] Joining separate fractions into one fraction ALWAYS drops the total value. Formula: x a 2 + y b 2 ≥ x + y ( a + b ) 2 Rule: Add the tops together, square it, and divide by the added bottoms. 2.5 Geometric Progressions & Series [ THE MULTIPLYING CHAIN ] A sequence where you multiply by the exact same multiplier to jump to the next term. [ FINITE SUM ] Adding up a specific, fixed number of multiplying steps. Formula 1: ∑ = a ( r n − 1 ) r − 1 Formula 2: ∑ = l r − a r − 1 Rule: a = first term , l = last term , r = multiplier , n = total terms. [ INFINITE SUM ] Adding up multiplying steps FOREVER. Only works if the numbers get smaller. Formula: ∑ = a 1 − r Rule: Only use this if the multiplier r is strictly between − 1 and 1. \begin{aligned}&\textbf{[ THE MULTIPLYING CHAIN ]} \\&\text{A sequence where you multiply by the exact same multiplier to jump to the next term.} \\\\&\textbf{[ FINITE SUM ]} \\&\text{Adding up a specific, fixed number of multiplying steps.} \\\\&\text{Formula 1:} \quad \sum = \frac{a(r^n - 1)}{r - 1} \\&\text{Formula 2:} \quad \sum = \frac{lr - a}{r - 1} \\&\text{Rule:} \quad a = \text{first term}, \,\, l = \text{last term}, \,\, r = \text{multiplier}, \,\, n = \text{total terms.} \\\\\\&\textbf{[ INFINITE SUM ]} \\&\text{Adding up multiplying steps FOREVER. Only works if the numbers get smaller.} \\\\&\text{Formula:} \quad \sum = \frac{a}{1 - r} \\&\text{Rule:} \quad \text{Only use this if the multiplier } r \text{ is strictly between } -1 \text{ and } 1.\end{aligned} [ THE MULTIPLYING CHAIN ] A sequence where you multiply by the exact same multiplier to jump to the next term. [ FINITE SUM ] Adding up a specific, fixed number of multiplying steps. Formula 1: ∑ = r − 1 a ( r n − 1 ) Formula 2: ∑ = r − 1 l r − a Rule: a = first term , l = last term , r = multiplier , n = total terms. [ INFINITE SUM ] Adding up multiplying steps FOREVER. Only works if the numbers get smaller. Formula: ∑ = 1 − r a Rule: Only use this if the multiplier r is strictly between − 1 and 1.
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