Trigonometry Formula Sheet A quick-reference sheet of essential trigonometry formulas. Covers trig ratios, unit circle values, identities, inverse functions, and the laws of sines and cosines. Each formula links to its full definition page.
Basic Trig Ratios sin θ = opposite hypotenuse \sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} sin θ = hypotenuse opposite cos θ = adjacent hypotenuse \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} cos θ = hypotenuse adjacent tan θ = opposite adjacent = sin θ cos θ \tan\theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{\sin\theta}{\cos\theta} tan θ = adjacent opposite = cos θ sin θ csc θ = 1 sin θ \csc\theta = \frac{1}{\sin\theta} csc θ = sin θ 1 sec θ = 1 cos θ \sec\theta = \frac{1}{\cos\theta} sec θ = cos θ 1 cot θ = 1 tan θ = cos θ sin θ \cot\theta = \frac{1}{\tan\theta} = \frac{\cos\theta}{\sin\theta} cot θ = tan θ 1 = sin θ cos θ
Pythagorean Identities sin 2 θ + cos 2 θ = 1 \sin^2\theta + \cos^2\theta = 1 sin 2 θ + cos 2 θ = 1 1 + tan 2 θ = sec 2 θ 1 + \tan^2\theta = \sec^2\theta 1 + tan 2 θ = sec 2 θ 1 + cot 2 θ = csc 2 θ 1 + \cot^2\theta = \csc^2\theta 1 + cot 2 θ = csc 2 θ Sum & Difference Formulas sin ( A ± B ) = sin A cos B ± cos A sin B \sin(A \pm B) = \sin A\cos B \pm \cos A\sin B sin ( A ± B ) = sin A cos B ± cos A sin B cos ( A ± B ) = cos A cos B ∓ sin A sin B \cos(A \pm B) = \cos A\cos B \mp \sin A\sin B cos ( A ± B ) = cos A cos B ∓ sin A sin B tan ( A + B ) = tan A + tan B 1 − tan A tan B \tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A\tan B} tan ( A + B ) = 1 − tan A tan B tan A + tan B Double & Half Angle sin 2 θ = 2 sin θ cos θ \sin 2\theta = 2\sin\theta\cos\theta sin 2 θ = 2 sin θ cos θ cos 2 θ = cos 2 θ − sin 2 θ \cos 2\theta = \cos^2\theta - \sin^2\theta cos 2 θ = cos 2 θ − sin 2 θ tan 2 θ = 2 tan θ 1 − tan 2 θ \tan 2\theta = \frac{2\tan\theta}{1-\tan^2\theta} tan 2 θ = 1 − tan 2 θ 2 tan θ sin θ 2 = ± 1 − cos θ 2 \sin\frac{\theta}{2} = \pm\sqrt{\frac{1-\cos\theta}{2}} sin 2 θ = ± 2 1 − cos θ cos θ 2 = ± 1 + cos θ 2 \cos\frac{\theta}{2} = \pm\sqrt{\frac{1+\cos\theta}{2}} cos 2 θ = ± 2 1 + cos θ Laws of Sines & Cosines a sin A = b sin B = c sin C \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} sin A a = sin B b = sin C c c 2 = a 2 + b 2 − 2 a b cos C c^2 = a^2 + b^2 - 2ab\cos C c 2 = a 2 + b 2 − 2 ab cos C A = 1 2 a b sin C A = \frac{1}{2}ab\sin C A = 2 1 ab sin C Unit Circle Key Values sin π 6 = 1 2 , cos π 6 = 3 2 \sin\frac{\pi}{6}=\frac{1}{2},\quad\cos\frac{\pi}{6}=\frac{\sqrt{3}}{2} sin 6 π = 2 1 , cos 6 π = 2 3 sin π 4 = 2 2 , cos π 4 = 2 2 \sin\frac{\pi}{4}=\frac{\sqrt{2}}{2},\quad\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2} sin 4 π = 2 2 , cos 4 π = 2 2 sin π 3 = 3 2 , cos π 3 = 1 2 \sin\frac{\pi}{3}=\frac{\sqrt{3}}{2},\quad\cos\frac{\pi}{3}=\frac{1}{2} sin 3 π = 2 3 , cos 3 π = 2 1
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