Trigonometric Facts

Pythagorean Identities

• cos²x + sin²x = 1
• 1 + tan²x = sec²x
• 1 + cot²x = csc²x

Addition Formulas

• cos(x + y) = cos(x)·cos(y) – sin(x)·sin(y); cos(x – y) = cos(x)·cos(y) + sin(x)·sin(y)
• sin(x + y) = sin(x)·cos(y) + cos(x)·sin(y); sin(x – y) = sin(x)·cos(y) – cos(x)·sin(y)

Double-Angle Formulas

• cos(2x) = cos²(x) – sin²(x)
• sin(2x) = 2 sin(x)·cos(x)

Half-Angle Formulas

• cos²(x) = (1 + cos(2x)) / 2
• sin²(x) = (1 – cos(2x)) / 2

Derivatives

• D_xsin(x) = cos(x); D_xcos(x) = –sin(x)
• D_xtan(x) = sec²(x); D_xcot(x) = –csc²(x)
• D_xsec(x) = sec(x)·tan(x); D_xcsc(x) = –csc(x)·cot(x)

Integrals

• ∫ sin(x) dx = –cos(x) + C; ∫ cos(x) dx = sin(x) + C
• ∫ tan(x) dx = ln | sec(x) | + C; ∫ cot(x) dx = –ln | csc(x) | + C
• ∫ sec(x) dx = ln | sec(x) + tan(x) | + C; ∫ csc(x) dx = –ln | csc(x) + cot(x) | + C

Inverse Functions

• D_xsin^–1(x) = 1 / (1 – x²)^1/2, |x| < 1; D_xcos^–1(x) = – 1 / (1 – x²)^1/2, |x| < 1
• D_xtan^–1(x) = 1 / (1 + x²); D_xcot^–1(x) = – 1 / (1 + x²)
• D_xsec^–1(x) = 1 / (|x|·(x² – 1)^1/2), |x| > 1; D_xcsc^–1(x) = – 1 / (|x|·(x² – 1)^1/2), |x| > 1