The trigonometric identity reference lists all major trig identities — Pythagorean, reciprocal, quotient, sum/difference, double angle, half angle, and product-to-sum — with a live search filter and numerical verification examples.
Trigonometric Identity Reference
How to Use the Trigonometric Identity Reference
Trigonometric identities are equations involving trig functions that hold true for all values of the variables. They're used to simplify expressions, solve equations, and prove other mathematical results in calculus, physics, and engineering.
Pythagorean Identities
The three Pythagorean identities all stem from sin²(x) + cos²(x) = 1 (the unit circle equation). Dividing by cos²(x) gives 1 + tan²(x) = sec²(x). Dividing by sin²(x) gives cot²(x) + 1 = csc²(x).
Sum and Difference Formulas
sin(a±b) = sin(a)cos(b) ± cos(a)sin(b). cos(a±b) = cos(a)cos(b) ∓ sin(a)sin(b). These enable calculating trig values for non-standard angles: sin(75°) = sin(45°+30°) = (√6+√2)/4 ≈ 0.966.
Double and Half Angle Formulas
Double angle: sin(2x) = 2sin(x)cos(x). cos(2x) = cos²(x)−sin²(x) = 2cos²(x)−1 = 1−2sin²(x). Half angle: sin(x/2) = ±√((1−cos(x))/2). These are derived from sum formulas with a = b = x.
FAQ
What are the three Pythagorean identities?
The three Pythagorean identities are: sin²(x) + cos²(x) = 1, 1 + tan²(x) = sec²(x), and 1 + cot²(x) = csc²(x). All three are derived from the Pythagorean theorem applied to a unit circle. The first is the most fundamental and is used to derive the other two.
What is the sum formula for sin(a+b)?
sin(a+b) = sin(a)cos(b) + cos(a)sin(b). Similarly, sin(a−b) = sin(a)cos(b) − cos(a)sin(b). For cosine: cos(a+b) = cos(a)cos(b) − sin(a)sin(b). These identities are essential for simplifying trig expressions and solving equations.
What are double angle formulas?
sin(2x) = 2sin(x)cos(x). cos(2x) = cos²(x)−sin²(x) = 2cos²(x)−1 = 1−2sin²(x). tan(2x) = 2tan(x)/(1−tan²(x)). These are derived from sum formulas with a=b=x.
Is this reference free?
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What is the SOHCAHTOA mnemonic?
SOHCAHTOA helps remember basic trig ratios in right triangles: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. For a 30-60-90 triangle: sin(30°)=0.5, cos(30°)=√3/2≈0.866, tan(30°)=1/√3≈0.577.