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The whole subject in your browser: 45 KaTeX-rendered identities across 16 topics, an interactive unit circle, a step-by-step triangle solver, tracked quizzes, and a real-world problem bank.

45

identities across 16 trigonometry topics

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SOHCAHTOA

\sin\theta = \dfrac{\text{opp}}{\text{hyp}}\quad \cos\theta = \dfrac{\text{adj}}{\text{hyp}}\quad \tan\theta = \dfrac{\text{opp}}{\text{adj}}

Mnemonic for the three core ratios in a right triangle.

Degrees ↔ Radians

1^{\circ} = \dfrac{\pi}{180}\text{ rad}\qquad 1\text{ rad} = \dfrac{180}{\pi}^{\!\!\circ}

Arc length

s = r\theta

θ must be in radians.

Sector area

A = \tfrac{1}{2} r^{2} \theta

θ in radians.

Pythagorean identity

\sin^{2}\theta + \cos^{2}\theta = 1

Secant form

1 + \tan^{2}\theta = \sec^{2}\theta

Cosecant form

1 + \cot^{2}\theta = \csc^{2}\theta

Secant

\sec\theta = \dfrac{1}{\cos\theta}

Cosecant

\csc\theta = \dfrac{1}{\sin\theta}

Cotangent

\cot\theta = \dfrac{1}{\tan\theta}

Tangent

\tan\theta = \dfrac{\sin\theta}{\cos\theta}

Cotangent

\cot\theta = \dfrac{\cos\theta}{\sin\theta}

Sine is odd

\sin(-\theta) = -\sin\theta

Cosine is even

\cos(-\theta) = \cos\theta

Tangent is odd

\tan(-\theta) = -\tan\theta

sin/cos

\sin(\tfrac{\pi}{2} - \theta) = \cos\theta

cos/sin

\cos(\tfrac{\pi}{2} - \theta) = \sin\theta

tan/cot

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

Sine

\sin(\alpha \pm \beta) = \sin\alpha \cos\beta \pm \cos\alpha \sin\beta

Sum & Difference

Cosine

\cos(\alpha \pm \beta) = \cos\alpha \cos\beta \mp \sin\alpha \sin\beta

Sum & Difference

Tangent

\tan(\alpha \pm \beta) = \dfrac{\tan\alpha \pm \tan\beta}{1 \mp \tan\alpha \tan\beta}

Sum & Difference

sin 2θ

\sin 2\theta = 2 \sin\theta \cos\theta

cos 2θ

\cos 2\theta = \cos^{2}\theta - \sin^{2}\theta = 2\cos^{2}\theta - 1 = 1 - 2\sin^{2}\theta

tan 2θ

\tan 2\theta = \dfrac{2\tan\theta}{1 - \tan^{2}\theta}

sin θ/2

\sin\tfrac{\theta}{2} = \pm\sqrt{\dfrac{1 - \cos\theta}{2}}

cos θ/2

\cos\tfrac{\theta}{2} = \pm\sqrt{\dfrac{1 + \cos\theta}{2}}

tan θ/2

\tan\tfrac{\theta}{2} = \dfrac{1 - \cos\theta}{\sin\theta} = \dfrac{\sin\theta}{1 + \cos\theta}

sin² θ

\sin^{2}\theta = \dfrac{1 - \cos 2\theta}{2}

Power Reduction

cos² θ

\cos^{2}\theta = \dfrac{1 + \cos 2\theta}{2}

Power Reduction

sin · cos

\sin\alpha \cos\beta = \tfrac{1}{2}[\sin(\alpha+\beta) + \sin(\alpha-\beta)]

cos · cos

\cos\alpha \cos\beta = \tfrac{1}{2}[\cos(\alpha-\beta) + \cos(\alpha+\beta)]

sin · sin

\sin\alpha \sin\beta = \tfrac{1}{2}[\cos(\alpha-\beta) - \cos(\alpha+\beta)]

sin A + sin B

\sin A + \sin B = 2 \sin\tfrac{A+B}{2} \cos\tfrac{A-B}{2}

sin A - sin B

\sin A - \sin B = 2 \cos\tfrac{A+B}{2} \sin\tfrac{A-B}{2}

cos A + cos B

\cos A + \cos B = 2 \cos\tfrac{A+B}{2} \cos\tfrac{A-B}{2}

cos A - cos B

\cos A - \cos B = -2 \sin\tfrac{A+B}{2} \sin\tfrac{A-B}{2}

Law of Sines

\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}

Circumradius form

\dfrac{a}{\sin A} = 2R

R = circumradius of the triangle.

For side a

a^{2} = b^{2} + c^{2} - 2bc \cos A

For angle A

\cos A = \dfrac{b^{2} + c^{2} - a^{2}}{2bc}

SAS area

\text{Area} = \tfrac{1}{2} a b \sin C

Heron's formula

\text{Area} = \sqrt{s(s-a)(s-b)(s-c)},\ \ s = \tfrac{a+b+c}{2}

arcsin range

\arcsin : [-1, 1] \to [-\tfrac{\pi}{2}, \tfrac{\pi}{2}]

arccos range

\arccos : [-1, 1] \to [0, \pi]

arctan range

\arctan : \mathbb{R} \to (-\tfrac{\pi}{2}, \tfrac{\pi}{2})