Trigonometric identities chart
The core trigonometric identities, grouped by type. These equations hold for every angle and are the tools for simplifying expressions, solving equations, and proving results throughout trigonometry and calculus.
Key identities
| Type | Identities |
|---|---|
| Pythagorean | sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = csc²θ |
| Reciprocal | cscθ = 1/sinθ; secθ = 1/cosθ; cotθ = 1/tanθ |
| Quotient | tanθ = sinθ/cosθ; cotθ = cosθ/sinθ |
| Even / odd | sin(−θ) = −sinθ; cos(−θ) = cosθ; tan(−θ) = −tanθ |
| Sum | sin(A+B) = sinA cosB + cosA sinB; cos(A+B) = cosA cosB − sinA sinB |
| Difference | sin(A−B) = sinA cosB − cosA sinB; cos(A−B) = cosA cosB + sinA sinB |
| Double angle | sin2θ = 2sinθcosθ; cos2θ = cos²θ − sin²θ |
The Pythagorean identity, sine squared plus cosine squared equals one, is the most used of all and comes straight from the unit circle. The reciprocal and quotient identities define the other four functions in terms of sine and cosine. The sum and double-angle formulas let you break complicated angles into simpler ones.
Need the values or circle formulas?
See the Unit Circle Chart and the Circle Formula Chart.
The Pythagorean identity
On the unit circle a point has coordinates cosine and sine, and the Pythagorean theorem on the radius gives cosine squared plus sine squared equals one. Dividing this identity through by cosine squared or sine squared produces the other two Pythagorean forms involving secant, tangent, cosecant, and cotangent. It is the foundation most other simplifications rest on.
Sum and double-angle formulas
The sum formulas express the sine and cosine of a combined angle using the sines and cosines of its parts. Setting both parts equal gives the double-angle formulas, which rewrite functions of twice an angle in terms of the single angle. These are essential for integrating trig functions and solving equations where angles combine.
FAQ
What is the most important trig identity?
The Pythagorean identity, sin²θ + cos²θ = 1. It holds for every angle and underlies most trigonometric simplification.
What are the reciprocal identities?
Cosecant is 1 over sine, secant is 1 over cosine, and cotangent is 1 over tangent. They define the three less-common functions.
What is the double angle formula for sine?
sin(2θ) = 2 sinθ cosθ. It rewrites the sine of a doubled angle using the single angle.