Trig identities cheat sheet

Almost every identity below can be rebuilt from the Pythagorean identity and the two sum formulas. Memorise those three and derive the rest when you need them.

Practise trigonometry

Reciprocal & quotient

$csc θ = \frac{1}{s i n θ}, sec θ = \frac{1}{c o s θ}, cot θ = \frac{1}{t a n θ}$
$tan θ = \frac{s i n θ}{c o s θ}$

Pythagorean

$sin^{2} θ + cos^{2} θ = 1$
$1 + tan^{2} θ = sec^{2} θ$
$1 + cot^{2} θ = csc^{2} θ$

Sum & difference

$sin (A \pm B) = sin A cos B \pm cos A sin B$
$cos (A \pm B) = cos A cos B \mp sin A sin B$
$tan (A \pm B) = \frac{t a n A \pm t a n B}{1 \mp t a n A t a n B}$

Double & half angle

$sin 2 θ = 2 sin θ cos θ$
$cos 2 θ = cos^{2} θ - sin^{2} θ = 1 - 2 sin^{2} θ$
$tan 2 θ = \frac{2 t a n θ}{1 - t a n ^{2} θ}$
$sin^{2} \frac{θ}{2} = \frac{1 - c o s θ}{2}$

Triangle laws

$\frac{a}{s i n A} = \frac{b}{s i n B} = \frac{c}{s i n C}$ Law of sines
$c^{2} = a^{2} + b^{2} - 2 ab cos C$ Law of cosines
$Area = \frac{1}{2} ab sin C$

Related calculators

Right Triangle Calculator