Trig Formula Reference Card

Quick-reference card for all key trigonometric formulas, identities, and values. Print-friendly layout for study sessions.

Unit Circle Values

Angle (deg) Angle (rad) \(\sin\theta\) \(\cos\theta\) \(\tan\theta\)

\(0^\circ\)

\(0\)

\(0\)

\(1\)

\(0\)

\(30^\circ\)

\(\frac{\pi}\{6}\)

\(\frac{1}\{2}\)

\(\frac{\sqrt{3}}\{2}\)

\(\frac{\sqrt{3}}\{3}\)

\(45^\circ\)

\(\frac{\pi}\{4}\)

\(\frac{\sqrt\{2}}\{2}\)

\(\frac{\sqrt\{2}}\{2}\)

\(1\)

\(60^\circ\)

\(\frac{\pi}\{3}\)

\(\frac{\sqrt{3}}\{2}\)

\(\frac{1}\{2}\)

\(\sqrt{3}\)

\(90^\circ\)

\(\frac{\pi}\{2}\)

\(1\)

\(0\)

undefined

\(120^\circ\)

\(\frac\{2\pi}\{3}\)

\(\frac{\sqrt{3}}\{2}\)

\(-\frac{1}\{2}\)

\(-\sqrt{3}\)

\(135^\circ\)

\(\frac\{3\pi}\{4}\)

\(\frac{\sqrt\{2}}\{2}\)

\(-\frac{\sqrt\{2}}\{2}\)

\(-1\)

\(150^\circ\)

\(\frac\{5\pi}\{6}\)

\(\frac{1}\{2}\)

\(-\frac{\sqrt{3}}\{2}\)

\(-\frac{\sqrt{3}}\{3}\)

\(180^\circ\)

\(\pi\)

\(0\)

\(-1\)

\(0\)

\(210^\circ\)

\(\frac\{7\pi}\{6}\)

\(-\frac{1}\{2}\)

\(-\frac{\sqrt{3}}\{2}\)

\(\frac{\sqrt{3}}\{3}\)

\(225^\circ\)

\(\frac\{5\pi}\{4}\)

\(-\frac{\sqrt\{2}}\{2}\)

\(-\frac{\sqrt\{2}}\{2}\)

\(1\)

\(240^\circ\)

\(\frac\{4\pi}\{3}\)

\(-\frac{\sqrt{3}}\{2}\)

\(-\frac{1}\{2}\)

\(\sqrt{3}\)

\(270^\circ\)

\(\frac\{3\pi}\{2}\)

\(-1\)

\(0\)

undefined

\(300^\circ\)

\(\frac\{5\pi}\{3}\)

\(-\frac{\sqrt{3}}\{2}\)

\(\frac{1}\{2}\)

\(-\sqrt{3}\)

\(315^\circ\)

\(\frac\{7\pi}\{4}\)

\(-\frac{\sqrt\{2}}\{2}\)

\(\frac{\sqrt\{2}}\{2}\)

\(-1\)

\(330^\circ\)

\(\frac\{11\pi}\{6}\)

\(-\frac{1}\{2}\)

\(\frac{\sqrt{3}}\{2}\)

\(-\frac{\sqrt{3}}\{3}\)

\(360^\circ\)

\(2\pi\)

\(0\)

\(1\)

\(0\)

Quadrant Signs (ASTC)

  • Q I (\(0^\circ\) to \(90^\circ\)): All positive

  • Q II (\(90^\circ\) to \(180^\circ\)): Sine positive

  • Q III (\(180^\circ\) to \(270^\circ\)): Tangent positive

  • Q IV (\(270^\circ\) to \(360^\circ\)): Cosine positive

Conversion Formulas

\[\text{Radians} = \text{Degrees} \times \frac{\pi}{180}\]
\[\text{Degrees} = \text{Radians} \times \frac{180}{\pi}\]
\[s = r\theta \quad \text{(arc length, } \theta \text{ in radians)}\]
\[A = \frac{1}{2}r^2\theta \quad \text{(sector area, } \theta \text{ in radians)}\]

Reciprocal Identities

\[\csc\theta = \frac{1}{\sin\theta} \qquad \sec\theta = \frac{1}{\cos\theta} \qquad \cot\theta = \frac{1}{\tan\theta}\]

Quotient Identities

\[\tan\theta = \frac{\sin\theta}{\cos\theta} \qquad \cot\theta = \frac{\cos\theta}{\sin\theta}\]

Pythagorean Identities

\[\sin^2\theta + \cos^2\theta = 1\]
\[1 + \tan^2\theta = \sec^2\theta\]
\[1 + \cot^2\theta = \csc^2\theta\]

Even-Odd Identities

\[\sin(-\theta) = -\sin\theta \qquad \cos(-\theta) = \cos\theta \qquad \tan(-\theta) = -\tan\theta\]

Cofunction Identities

\[\sin\left(\frac{\pi}{2} - \theta\right) = \cos\theta \qquad \cos\left(\frac{\pi}{2} - \theta\right) = \sin\theta\]
\[\tan\left(\frac{\pi}{2} - \theta\right) = \cot\theta \qquad \cot\left(\frac{\pi}{2} - \theta\right) = \tan\theta\]

Sum and Difference Formulas

\[\sin(\alpha \pm \beta) = \sin\alpha\cos\beta \pm \cos\alpha\sin\beta\]
\[\cos(\alpha \pm \beta) = \cos\alpha\cos\beta \mp \sin\alpha\sin\beta\]
\[\tan(\alpha \pm \beta) = \frac{\tan\alpha \pm \tan\beta}{1 \mp \tan\alpha\tan\beta}\]

Double Angle Formulas

\[\sin(2\theta) = 2\sin\theta\cos\theta\]
\[\cos(2\theta) = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta\]
\[\tan(2\theta) = \frac{2\tan\theta}{1 - \tan^2\theta}\]

Half Angle Formulas

\[\sin\frac{\theta}{2} = \pm\sqrt{\frac{1 - \cos\theta}{2}}\]
\[\cos\frac{\theta}{2} = \pm\sqrt{\frac{1 + \cos\theta}{2}}\]
\[\tan\frac{\theta}{2} = \frac{1 - \cos\theta}{\sin\theta} = \frac{\sin\theta}{1 + \cos\theta}\]

Power-Reducing Formulas

\[\sin^2\theta = \frac{1 - \cos(2\theta)}{2} \qquad \cos^2\theta = \frac{1 + \cos(2\theta)}{2}\]

Product-to-Sum Formulas

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

Sum-to-Product Formulas

\[\sin\alpha + \sin\beta = 2\sin\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}\]
\[\cos\alpha + \cos\beta = 2\cos\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}\]

Law of Sines

\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]

Law of Cosines

\[c^2 = a^2 + b^2 - 2ab\cos C\]

Area Formulas

\[A = \frac{1}{2}ab\sin C\]

Heron’s formula:

\[A = \sqrt{s(s-a)(s-b)(s-c)} \quad \text{where } s = \frac{a+b+c}{2}\]

Inverse Trig Function Ranges

Function Domain Range

\(\arcsin(x)\)

\(\lbrack -1, 1 \rbrack\)

\(\lbrack -\frac{\pi\}{2\}, \frac{\pi\}{2\} \rbrack\)

\(\arccos(x)\)

\(\lbrack -1, 1 \rbrack\)

\(\lbrack 0, \pi \rbrack\)

\(\arctan(x)\)

\((-\infty, \infty)\)

\((-\frac{\pi\}{2\}, \frac{\pi\}{2\})\)

\(\operatorname\{arccot\}(x)\)

\((-\infty, \infty)\)

\((0, \pi)\)

\(\operatorname\{arcsec\}(x)\)

\((-\infty,-1\rbrack \cup \lbrack 1,\infty)\)

\(\lbrack 0, \frac{\pi\}{2\}) \cup (\frac{\pi\}{2\}, \pi \rbrack\)

\(\operatorname\{arccsc\}(x)\)

\((-\infty,-1\rbrack \cup \lbrack 1,\infty)\)

\(\lbrack -\frac{\pi\}{2\}, 0) \cup (0, \frac{\pi\}{2\} \rbrack\)

Polar Coordinate Conversions

\[x = r\cos\theta \qquad y = r\sin\theta\]
\[r = \sqrt{x^2 + y^2} \qquad \theta = \arctan\frac{y}{x}\]

Complex Numbers in Polar Form

\[z = r(\cos\theta + i\sin\theta) = re^{i\theta}\]

De Moivre’s Theorem:

\[z^n = r^n(\cos(n\theta) + i\sin(n\theta))\]

Conic Sections in Polar Form

\[r = \frac{ed}{1 \pm e\cos\theta} \quad \text{or} \quad r = \frac{ed}{1 \pm e\sin\theta}\]

  • \(e < 1\): ellipse

  • \(e = 1\): parabola

  • \(e > 1\): hyperbola