Best for
Fast trig lookup
Use the table for common angles, exact trig values, and identities used in engineering calculations.
Engineering formula reference
Trigonometric Identities Formula Sheet
Search trigonometric values and identities used in engineering math, vectors, waves, signals, mechanics, machine design, and geometry calculations.
Engineering math
Search trigonometric identities formula sheet
Filter the formula table by shape, variable, symbol, expression, or engineering use case.
Angle mode
Degrees vs radians
Check whether your calculator, code, or spreadsheet expects degrees or radians before substituting angles.
Formula reference
Common trigonometric values and identities
| Topic | Identity or value | Use |
|---|---|---|
| 0 degrees | sin 0 = 0, cos 0 = 1, tan 0 = 0 | Unit circle starting point |
| 30 degrees | sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3 | Common triangle angle |
| 45 degrees | sin 45° = √2/2, cos 45° = √2/2, tan 45° = 1 | Equal-leg right triangle |
| 60 degrees | sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3 | Common triangle angle |
| 90 degrees | sin 90° = 1, cos 90° = 0, tan 90° undefined | Tangent undefined |
| Pythagorean | sin2θ + cos2θ = 1 | Simplification and checks |
| Quotient | tanθ = sinθ / cosθ | Convert between trig functions |
| Double angle sine | sin(2θ) = 2sinθcosθ | Waves, vectors, and transformations |
| Double angle cosine | cos(2θ) = cos2θ - sin2θ | Expression simplification |
| Angle sum sine | sin(a + b) = sin a cos b + cos a sin b | Phase and vector calculations |
| Angle sum cosine | cos(a + b) = cos a cos b - sin a sin b | Phase and coordinate transforms |
How to use
Practical engineering checks
Degrees to radians
rad = deg × π / 180
Use radians for most calculus, programming, and engineering math libraries.
Radians to degrees
deg = rad × 180 / π
Use degrees for many drawing, geometry, and shop-floor angle references.
Pythagorean identity
sin2θ + cos2θ = 1
Useful for simplifying expressions and checking trig values.
Tangent
tanθ = sinθ / cosθ
Tangent is undefined when cos theta is zero.
Definitions
Symbols and variables explained
θ
Theta, the angle used in the trigonometric expression.
sinθ
Opposite side divided by hypotenuse in a right triangle.
cosθ
Adjacent side divided by hypotenuse in a right triangle.
tanθ
Opposite side divided by adjacent side, or sin theta divided by cos theta.
FAQ
Trigonometric Identities Formula Sheet questions
How do I use this formula sheet?
Find the shape or identity, confirm each variable definition, keep units consistent, then substitute values into the formula.
Are these formulas exact?
The basic geometry formulas are exact for ideal shapes. Rows marked approximate depend on simplified geometry, thin-wall assumptions, or ignoring radii and fillets.
Can I use these formulas for final engineering design?
Use them as calculation references. Final design should also check loads, materials, tolerances, safety factors, and any governing standards.
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