Trigonometric Identities and Vector Formulas

Pythagorean identity involving sin and cos

sin²x + cos²x = 1

Identity for 1 + tan²x

1 + tan²x = sec²x

Identity for 1 + cot²x

1 + cot²x = csc²x

tan(x) in terms of sin and cos

tan(x) = sin(x)/cos(x)

sec(x) in terms of cos(x)

sec(x) = 1/cos(x)

sin(a ± b)

sin(a ± b) = sin(a)cos(b) ± cos(a)sin(b)

cos(a ± b)

cos(a ± b) = cos(a)cos(b) ∓ sin(a)sin(b)

tan(a ± b)

tan(a ± b) = (tan(a) ± tan(b)) / (1 ∓ tan(a)tan(b))

sin(2x)

sin(2x) = 2sin(x)cos(x)

cos(2x)

cos(2x) = cos²x − sin²x

Alternate cos(2x) in terms of cos(x)

cos(2x) = 2cos²x − 1

Alternate cos(2x) in terms of sin(x)

cos(2x) = 1 − 2sin²x

Domain and range of sin⁻¹(x)

Domain: [−1,1], Range: [−π/2, π/2]

Domain and range of cos⁻¹(x)

Domain: [−1,1], Range: [0, π]

Domain and range of tan⁻¹(x)

Domain: (−∞, ∞), Range: (−π/2, π/2)

Formula for component form of vector PQ

PQ = ⟨x₂−x₁, y₂−y₁⟩

Magnitude of vector ⟨a, b⟩

|v| = √(a² + b²)

Write vector v = |v| in direction θ in i/j form

v = |v|cosθ i + |v|sinθ j

Unit vectors i and j

i = ⟨1, 0⟩, j = ⟨0, 1⟩