Trig – Tricks with TEA and Toast (T⁴)

A comprehensive set of vocabulary flashcards summarizing key trigonometric identities, reciprocal and quotient relations, cofunction and odd-even properties, Pythagorean, sum/difference, and double-angle formulas, along with triangle laws and area formulas.

sec²θ

Equals 1 + tan²θ (Pythagorean identity).

tan²θ in terms of sec²θ

tan²θ = sec²θ − 1.

tanθ in terms of sin and cos

tanθ = sinθ / cosθ.

cotθ in terms of sin and cos

cotθ = cosθ / sinθ.

sin²θ in terms of cos²θ

sin²θ = 1 − cos²θ.

cos²θ in terms of sin²θ

cos²θ = 1 − sin²θ.

cot²θ in terms of csc²θ

cot²θ = csc²θ − 1.

csc²θ in terms of cot²θ

csc²θ = 1 + cot²θ.

sinθ reciprocal identity

sinθ = 1 / cscθ.

cosθ reciprocal identity

cosθ = 1 / secθ.

secθ reciprocal identity

secθ = 1 / cosθ.

cscθ reciprocal identity

cscθ = 1 / sinθ.

cotθ reciprocal identity

cotθ = 1 / tanθ.

tanθ quotient identity

tanθ = sinθ / cosθ.

cotθ quotient identity

cotθ = cosθ / sinθ.

cos(−θ)

cos(−θ) = cosθ (even function).

sin(−θ)

sin(−θ) = −sinθ (odd function).

csc(−θ)

csc(−θ) = −cscθ (odd function).

tan(−θ)

tan(−θ) = −tanθ (odd function).

cot(−θ)

cot(−θ) = −cotθ (odd function).

Only positive for cos and sec when θ → −θ

cos(−θ) and sec(−θ) remain positive, others change sign.

sinθ cofunction identity

sinθ = cos(π/2 − θ).

cosθ cofunction identity

cosθ = sin(π/2 − θ).

tanθ cofunction identity

tanθ = cot(π/2 − θ).

cscθ cofunction identity

cscθ = sec(π/2 − θ).

cotθ cofunction identity

cotθ = tan(π/2 − θ).

sin²θ + cos²θ

Equals 1 (fundamental Pythagorean identity).

1 + tan²θ

Equals sec²θ (Pythagorean identity).

1 + cot²θ

Equals csc²θ (Pythagorean identity).

sin(A + B)

sinA cosB + cosA sinB (sum identity).

sin(A − B)

sinA cosB − cosA sinB (difference identity).

cos(A + B)

cosA cosB − sinA sinB (sum identity).

cos(A − B)

cosA cosB + sinA sinB (difference identity).

sin 2θ

2 sinθ cosθ (double-angle identity).

cos 2θ (primary form)

cos²θ − sin²θ.

cos 2θ (in terms of cos)

2 cos²θ − 1.

cos 2θ (in terms of sin)

1 − 2 sin²θ.

tan 2θ

2 tanθ / (1 − tan²θ) (double-angle identity).

sec²α − tan²α

Equals 1.

csc²α − cot²α

Equals 1.

(secα + tanα)(secα − tanα)

Equals 1 (difference of squares).

(cscα + cotα)(cscα − cotα)

Equals 1 (difference of squares).

If (secα + tanα) = μ

Then (secα − tanα) = 1/μ.

If (cscα − cotα) = ω

Then (cscα + cotα) = 1/ω.

sin(π/2 − x)

Equals cos x.

cos(π/2 − x)

Equals sin x.

tan(π/2 − x)

Equals cot x.

Slope of a line

tanθ = RISE / RUN = opposite / adjacent.

SOH CAH TOA

Mnemonic: sin = opp/hyp, cos = adj/hyp, tan = opp/adj.

Law of Sines

a/sinA = b/sinB = c/sinC.

Law of Cosines

c² = a² + b² − 2ab cosC (and cyclic variants).

Heron’s Formula

Area = √[s(s−a)(s−b)(s−c)], where s = (a + b + c)/2.

Area of a triangle using sine

Area = ½ ab sinC.