In This Lesson Fundamental Identities Sum & Difference Formulas Double & Half-Angle Formulas 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 Solving Trig Equations Inverse Trig Functions Fundamental Identities Pythagorean: Reciprocal: csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θQuotient: tan θ = sin θ/cos θ, cot θ = cos θ/sin θ
All three Pythagorean identities derive from sin²θ + cos²θ = 1 (from the unit circle ). They're essential for simplifying expressions in integration .
Sum & Difference Formulas
sin(A ± B) = sin A cos B ± cos A sin B
These let you expand trig functions of sums — crucial for Fourier analysis , deriving trig derivatives , and signal processing .
Double & Half-Angle Formulas Double angle: Half angle:
The half-angle formulas (also called power-reduction formulas) are essential for integrating sin²x and cos²x .
Solving Trig Equations Strategy: use identities to reduce to a single trig function, then solve like an algebraic equation . Remember that trig functions are periodic, so there are infinitely many solutions.
Example: Solve 2sin²x − sin x − 1 = 0 Let u = sin x: 2u² − u − 1 = 0
Factor : (2u + 1)(u − 1) = 0
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sin x = −1/2: x = 7π/6 + 2kπ or x = 11π/6 + 2kπ
sin x = 1: x = π/2 + 2kπ
Inverse Trig Functions Since trig functions aren't one-to-one, we restrict their domains to define inverses:
sin⁻¹(x): range [−π/2, π/2]
Inverse trig functions appear in integration (∫ dx/√(1−x²) = sin⁻¹x + C) and in differentiation (d/dx sin⁻¹x = 1/√(1−x²)).