In This Lesson Fundamental Identities Sum & Difference Formulas 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 Double & Half-Angle Formulas Solving Trig Equations 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 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
All three Pythagorean identities derive from sin²θ + cos²θ = 1 (from the unit circle ). They're essential for simplifying expressions in integration .
Sum & Difference Formulas 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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 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 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 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]
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)).