IP331.com | Online Tools
AlgebraGeometryTrigonometryMatricesFunctionsNumber TheoryBase ConverterElectronicsStatisticsLife & FinanceSimulators
Home›Trigonometry›Double Angle Formula Calculator

Double Angle Formula Calculator

Enter an angle to compute sin(2A), cos(2A), and tan(2A) with step-by-step derivation

Angle A (degrees)

Double Angle Formulas

sin(2A) = 2 sinA cosA
cos(2A) = cos^2A - sin^2A = 2cos^2A - 1 = 1 - 2sin^2A
tan(2A) = 2 tanA / (1 - tan^2A)

Double-angle formulas are among the most frequently used trigonometric identities. They allow you to express trigonometric functions of twice an angle in terms of the original angle.

⚠ For tan(2A), if cos(2A) = 0 or tanA = +/- 1, tan(2A) is undefined (vertical asymptote).

What Are Double-Angle Formulas?

Double-angle formulas are derived from the sum formulas by setting both angles equal. They are fundamental identities in trigonometry used extensively in calculus, physics, and engineering.

Sine Double

sin(2A) = 2 sinA cosA. Derived from sin(A+A) = sinAcosA + cosAsinA.

Cosine Triple Form

cos(2A) has three forms: cos^2-sin^2, 2cos^2-1, 1-2sin^2. Each is useful in different contexts.

Tangent Double

tan(2A) = 2tanA/(1-tan^2A). Undefined when tanA = +/-1 (A = 45 or 135 degrees).

Applications

Used in projectile motion (range formula), wave interference, AC circuits, and integration substitution.

Teaching Example: A = 30 degrees. sin30 = 0.5, cos30 = 0.866. sin(60) = 2x0.5x0.866 = 0.866. cos(60) = 0.866^2 - 0.5^2 = 0.75 - 0.25 = 0.5. tan(60) = 2x0.577/(1-0.333) = 1.155/0.667 = 1.732.

Applications

Projectile Motion Wave Physics Calculus AC Circuits Signal Processing Trig Proofs

FAQs about Double-Angle Formulas

What are double-angle formulas?▼
sin(2A) = 2sinAcosA, cos(2A) = cos^2A - sin^2A, tan(2A) = 2tanA/(1-tan^2A). Derived from sum formulas.
What is sin(2A) when A=30?▼
sin(60) = 2 x sin30 x cos30 = 2 x 0.5 x 0.866 = 0.866. The exact value is sqrt(3)/2.
What are the three forms of cos(2A)?▼
cos(2A) = cos^2A - sin^2A (basic), = 2cos^2A - 1, = 1 - 2sin^2A. Choose the form that simplifies your problem.
When is tan(2A) undefined?▼
tan(2A) is undefined when tanA = +/-1, which occurs at A = 45 and 135 degrees. At these points, 2A = 90 or 270 degrees where tan is undefined.

More Trigonometry Tools

Algebra Geometry Trigonometry Matrices Functions Number Theory Base Converter Electronics Statistics Life & Finance Simulators

Free online calculators and tools covering mathematics, unit conversion, text processing, and daily life. Accurate, fast, mobile-friendly, and completely free to use.

Home Sitemap About Contact

© 2026 IP331.com - Free Online Tools. All rights reserved.

Privacy Policy | Cookie Policy | Terms of Use | Disclaimer