In This Lesson Fundamental Identities Sum & Difference Formulas Double & Half-Angle Formulas 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 Solving Trig Equations Inverse Trig Functions 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 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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 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 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.
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)).
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