In This Lesson Fundamental Identities Sum & Difference Formulas Double & Half-Angle Formulas Solving Trig Equations Inverse Trig Functions 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 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 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 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
u = −1/2 or u = 1
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)).
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