Trigonometry

At a glance 12 formulas

1. 1Standard values

   $0^\circ$

   $0,\ 1,\ 0$

   $30^\circ = \tfrac{\pi}{6}$

   $\tfrac{1}{2},\ \tfrac{\sqrt{3}}{2},\ \tfrac{1}{\sqrt{3}}$

   $45^\circ = \tfrac{\pi}{4}$

   $\tfrac{1}{\sqrt{2}},\ \tfrac{1}{\sqrt{2}},\ 1$

   $60^\circ = \tfrac{\pi}{3}$

   $\tfrac{\sqrt{3}}{2},\ \tfrac{1}{2},\ \sqrt{3}$

   $90^\circ = \tfrac{\pi}{2}$

   $1,\ 0,\ \text{undefined}$

2. 2Squares add to 1$\sin^2\theta + \cos^2\theta = 1$$1 + \tan^2\theta = \sec^2\theta,\quad 1 + \cot^2\theta = \operatorname{cosec}^2\theta$
3. 3Complementary angles$\sin(90^\circ - \theta) = \cos\theta$$\tan(90^\circ - \theta) = \cot\theta,\ \text{so } \tan\theta\tan(90^\circ - \theta) = 1$
4. 4Signs by quadrant$\sin(180^\circ - \theta) = \sin\theta,\ \cos(180^\circ - \theta) = -\cos\theta$$\text{positive: I all, II sin, III tan, IV cos}$
5. 5Sum and difference$\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B$$\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B$
6. 6Tan of a sum$\tan(A + B) = \dfrac{\tan A + \tan B}{1 - \tan A\tan B}$$A + B = 45^\circ \Rightarrow (1 + \tan A)(1 + \tan B) = 2$
7. 7Double angle$\sin 2\theta = 2\sin\theta\cos\theta$$\cos 2\theta = \cos^2\theta - \sin^2\theta = 1 - 2\sin^2\theta$
8. 8Greatest and least value$-\sqrt{a^2 + b^2} \le a\sin x + b\cos x \le \sqrt{a^2 + b^2}$
9. 9Quadratic in one functionTurn $\sin^2 x$ into $1 - \cos^2 x$ (or back) to get a quadratic in one function; keep only values in $[-1, 1]$, and check the ends if the vertex falls outside.
10. 10Square and add$(\cos\alpha + \cos\beta)^2 + (\sin\alpha - \sin\beta)^2 = 2 + 2\cos(\alpha + \beta)$$(\cos\alpha + \cos\beta)^2 + (\sin\alpha + \sin\beta)^2 = 2 + 2\cos(\alpha - \beta)$
11. 11Doubling cosines$\cos A\cos 2A\cos 4A = \dfrac{\sin 8A}{8\sin A}$$\text{times } \sin A\text{, then } \sin 2\theta \text{ thrice}$
12. 12Counting solutionsIn $0 \le x < 2\pi$, $\sin x = k$ has two solutions if $-1 < k < 1$ and one if $k = \pm 1$. Factor out $\cos x$ instead of dividing by it, or its roots are lost.