In This Lesson Fundamental Identities Sum & Difference Formulas 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 Double & Half-Angle Formulas Solving Trig Equations Inverse Trig Functions Fundamental Identities Pythagorean:
sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = csc²θ Reciprocal: csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ Quotient: tan θ = sin θ/cos θ, cot θ = cos θ/sin θ
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sin(A ± B) = sin A cos B ± cos A sin B
cos(A ± B) = cos A cos B ∓ sin A sin B
tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan 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:
sin 2θ = 2 sin θ cos θ
cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ
tan 2θ = 2 tan θ / (1 − tan²θ) Half angle:
sin²(θ/2) = (1 − cos θ)/2
cos²(θ/2) = (1 + cos θ)/2
The half-angle formulas (also called power-reduction formulas) are essential for integrating sin²x and cos²x .
Solving Trig Equations 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 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.
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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 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 Since trig functions aren't one-to-one, we restrict their domains to define inverses:
sin⁻¹(x): range [−π/2, π/2]
cos⁻¹(x): range [0, π]
tan⁻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)).