I. Trigonometric Formulas for Special Related Arcs
1. Special Related Arcs
1.1 Two opposite arcs: ($\alpha$ and $-\alpha$)
- $\cos(-\alpha) = \cos\alpha$
- $\sin(-\alpha) = -\sin\alpha$
- $\tan(-\alpha) = -\tan\alpha$
- $\cot(-\alpha) = -\cot\alpha$
1.2 Two supplementary arcs: ($\alpha$ and $\pi - \alpha$)
- $\sin(\pi - \alpha) = \sin\alpha$
- $\cos(\pi - \alpha) = -\cos\alpha$
- $\tan(\pi - \alpha) = -\tan\alpha$
- $\cot(\pi - \alpha) = -\cot\alpha$
1.3 Two complementary arcs: ($\alpha$ and $\dfrac{\pi}{2} - \alpha$)
- $\sin\!\left(\dfrac{\pi}{2} - \alpha\right) = \cos\alpha$ $\qquad$ $\cos\!\left(\dfrac{\pi}{2} - \alpha\right) = \sin\alpha$
- $\tan\!\left(\dfrac{\pi}{2} - \alpha\right) = \cot\alpha$ $\qquad$ $\cot\!\left(\dfrac{\pi}{2} - \alpha\right) = \tan\alpha$
1.4 Two arcs differing by $\pi$: ($\alpha$ and $\pi + \alpha$)
- $\sin(\pi + \alpha) = -\sin\alpha$
- $\cos(\pi + \alpha) = -\cos\alpha$
- $\tan(\pi + \alpha) = \tan\alpha$
- $\cot(\pi + \alpha) = \cot\alpha$
1.5 Arcs differing by $\dfrac{\pi}{2}$:
- $\cos\!\left(\dfrac{\pi}{2} + x\right) = -\sin x$; $\quad$ $\sin\!\left(\dfrac{\pi}{2} + x\right) = \cos x$;
- Note: cosine changes; sine supplements; complementary crosses; differing by $\pi$: tan, cot unchanged
II. Basic Trigonometric Formulas and Addition Formulas
2. Basic Trigonometric Formulas
- $\sin^2 x + \cos^2 x = 1$ $\qquad\qquad$ $\dfrac{1}{\cos^2 x} = 1 + \tan^2 x$
- $\dfrac{1}{\sin^2 x} = 1 + \cot^2 x$ $\qquad\qquad$ $\tan x . \cot x = 1$
- $\tan x = \dfrac{\sin x}{\cos x}$ $\qquad\qquad$ $\cot x = \dfrac{\cos x}{\sin x}$
- $\sin(a \pm b) = \sin a.\cos b \pm \cos a.\sin b$
- $\cos(a \pm b) = \cos a.\cos b \mp \sin a.\sin b$
- $\tan(a \pm b) = \dfrac{\tan a \pm \tan b}{1 \mp \tan a.\tan b}$
III. Double Angle, Triple Angle and Power-Reduction Formulas
4. Multiplication Formulas
4.1 Double Angle Formulas
- $\sin 2a = 2\sin a\cos a$
- $\cos 2a = \cos^2 a - \sin^2 a = 2\cos^2 a - 1 = 1 - 2\sin^2 a$
- $\tan 2a = \dfrac{2\tan a}{1 - \tan^2 a}$
4.2 Triple Angle Formulas
- $\sin 3a = 3\sin a - 4\sin^3 a$
- $\cos 3a = 4\cos^3 a - 3\cos a$
- $\tan 3a = \dfrac{3\tan a - \tan^3 a}{1 - 3\tan^2 a}$
5. Power-Reduction Formulas
- $\sin^2 a = \dfrac{1 - \cos 2a}{2}$ $\qquad\qquad$ $\cos^2 a = \dfrac{1 + \cos 2a}{2}$
- $\sin^3 a = \dfrac{3\sin a - \sin 3a}{4}$ $\qquad\qquad$ $\cos^3 a = \dfrac{3\cos a + \cos 3a}{4}$
IV. Sum-to-Product and Product-to-Sum Formulas
6. Sum-to-Product Formulas
- $\cos a + \cos b = 2\cos\dfrac{a+b}{2}\cos\dfrac{a-b}{2}$
- $\cos a - \cos b = -2\sin\dfrac{a+b}{2}\sin\dfrac{a-b}{2}$
- $\sin a + \sin b = 2\sin\dfrac{a+b}{2}\cos\dfrac{a-b}{2}$
- $\sin a - \sin b = 2\cos\dfrac{a+b}{2}\sin\dfrac{a-b}{2}$
7. Product-to-Sum Formulas
- $\cos a.\cos b = \dfrac{1}{2}\left[\cos(a+b) + \cos(a-b)\right]$
- $\sin a.\sin b = -\dfrac{1}{2}\left[\cos(a+b) - \cos(a-b)\right]$
- $\sin a.\cos b = \dfrac{1}{2}\left[\sin(a+b) + \sin(a-b)\right]$
V. Solution Formulas for Basic Trigonometric Equations
Basic Knowledge- $\sin u = \sin v \Leftrightarrow \left[ {\begin{array}{{20}{l}} {u = v + k2\pi }\\ {u = \pi - v + k2\pi } \end{array}} \right.$
- $\cos u = \cos v \Leftrightarrow \left[ {\begin{array}{{20}{l}} {u = v + k2\pi }\\ {u = - v + k2\pi } \end{array}} \right.$
- $\tan u = \tan v \Leftrightarrow u = v + k\pi$
- $\cot u = \cot v \Leftrightarrow u = v + k\pi$
- $\sin u = 0 \Leftrightarrow u = k\pi$ $\qquad\qquad$ $\cos u = 0 \Leftrightarrow u = \dfrac{\pi}{2} + k\pi$
- $\sin u = 1 \Leftrightarrow u = \dfrac{\pi}{2} + k2\pi$ $\qquad\qquad$ $\cos u = 1 \Leftrightarrow u = k2\pi$
- $\sin u = -1 \Leftrightarrow u = -\dfrac{\pi}{2} + k2\pi$ $\qquad\qquad$ $\cos u = -1 \Leftrightarrow u = \pi + k2\pi$
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