Right-triangle ratios
$$\sin\theta=\frac{\text{opp}}{\text{hyp}},\quad\cos\theta=\frac{\text{adj}}{\text{hyp}},\quad\tan\theta=\frac{\text{opp}}{\text{adj}}$$
Connect an acute angle to ratios of right-triangle sides.
High schoolTrigonometry formulas, grouped by topic.
18 trigonometry formulas across 7 topics, free and without an account. Trigonometry formulas connect angles to lengths and to periodic behavior. The right-triangle ratios and the Pythagorean identities are the foundation; the sum, difference, double-angle, and half-angle identities let you rewrite an expression until it becomes something you can evaluate. Angle mode matters throughout: a formula stated in radians will quietly return nonsense from a calculator left in degrees.
Every trigonometry formula on one page
Each entry links to its own page, where the symbols are defined, the conditions are stated, and the formula is substituted through a worked example. Levels covered here: High school, AP, University.
Right triangles
1 formula in right triangles.
Identities
5 formulas in identities.
Reciprocal identities
$$\csc\theta=\frac1{\sin\theta},\quad\sec\theta=\frac1{\cos\theta},\quad\cot\theta=\frac1{\tan\theta}$$
Define cosecant, secant, and cotangent as reciprocals.
High school · APPythagorean identities
$$\sin^2\theta+\cos^2\theta=1,\quad1+\tan^2\theta=\sec^2\theta,\quad1+\cot^2\theta=\csc^2\theta$$
Relate squared trigonometric functions through the unit circle.
High school · APQuotient identities
$$\tan\theta=\frac{\sin\theta}{\cos\theta},\qquad\cot\theta=\frac{\cos\theta}{\sin\theta}$$
Express tangent and cotangent using sine and cosine.
High schoolCofunction identities
$$\sin\left(\frac\pi2-\theta\right)=\cos\theta,\qquad\cos\left(\frac\pi2-\theta\right)=\sin\theta$$
Connect complementary angles and paired trigonometric functions.
High schoolProduct-to-sum identities
$$2\sin A\cos B=\sin(A+B)+\sin(A-B)$$
Rewrite a product of trigonometric functions as a sum.
AP · UniversityAngle formulas
5 formulas in angle formulas.
Sine sum and difference
$$\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B$$
Find sine of a combined angle.
High school · APCosine sum and difference
$$\cos(A\pm B)=\cos A\cos B\mp\sin A\sin B$$
Find cosine of a combined angle.
High school · APTangent sum and difference
$$\tan(A\pm B)=\frac{\tan A\pm\tan B}{1\mp\tan A\tan B}$$
Find tangent of a combined angle.
High school · APDouble-angle identities
$$\sin2\theta=2\sin\theta\cos\theta,\quad\cos2\theta=\cos^2\theta-\sin^2\theta$$
Rewrite trigonometric functions of twice an angle.
High school · APHalf-angle identities
$$\sin^2\frac{\theta}{2}=\frac{1-\cos\theta}{2},\quad\cos^2\frac{\theta}{2}=\frac{1+\cos\theta}{2}$$
Connect half angles to cosine of the original angle.
High school · APTriangles
3 formulas in triangles.
Law of sines
$$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$$
Solve a non-right triangle using opposite side-angle pairs.
High schoolLaw of cosines
$$c^2=a^2+b^2-2ab\cos C$$
Solve a triangle from SSS or SAS information.
High schoolTriangle area with sine
$$A=\frac12ab\sin C$$
Find area from two sides and their included angle.
High schoolAngles
1 formula in angles.
Degrees and radians
$$\theta_{\rm rad}=\theta_{\rm deg}\frac{\pi}{180},\qquad\theta_{\rm deg}=\theta_{\rm rad}\frac{180}{\pi}$$
Convert the two common angle units.
High schoolGraphs
2 formulas in graphs.
Sinusoidal model
$$y=A\sin(B(x-C))+D$$
Model periodic behavior with amplitude, period, shift, and midline.
High school · APSinusoid amplitude and period
$$y=A\sin(B(x-C))+D,\quad \text{amplitude}=|A|,\quad \text{period}=\frac{2\pi}{|B|}$$
Read the shape and transformations of a sinusoidal model.
High school · APUnit circle
1 formula in unit circle.
Unit-circle coordinates
$$(x,y)=(\cos\theta,\sin\theta),\qquad\tan\theta=\frac yx$$
Read trigonometric values from a point on the unit circle.
High school · APKeep going with trigonometry
A formula sheet is a reference, not a method. Use the subject guide to see where these relationships come from, a calculator to check a substitution you are unsure about, and practice questions to find out whether you can actually retrieve them.
- Study trigonometry — the subject hub with lessons, worked methods, and common mistakes.
- Trigonometry calculator — enter your own values and compare the result with your working.
- Precalculus Practice — retrieval practice across every topic listed above.
- Math formula library — search all 215 formulas by name, symbol, or condition.
Formulas in the other subjects
Mathematics does not stop at a subject boundary, and neither do its formulas. Each subject page below lists its own relationships, conditions, and worked examples.
Arithmetic
Review the core arithmetic relationships, their restrictions, and worked substitutions.
Algebra
Review the core algebra relationships, their restrictions, and worked substitutions.
Geometry
Review the core geometry relationships, their restrictions, and worked substitutions.
Precalculus
Review the core precalculus relationships, their restrictions, and worked substitutions.
Calculus
Review the core calculus relationships, their restrictions, and worked substitutions.
Statistics
Review the core statistics relationships, their restrictions, and worked substitutions.
Probability
Review the core probability relationships, their restrictions, and worked substitutions.
Linear Algebra
Review the core linear algebra relationships, their restrictions, and worked substitutions.
Differential Equations
Review the core differential equations relationships, their restrictions, and worked substitutions.
Discrete Math
Review the core discrete math relationships, their restrictions, and worked substitutions.