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
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
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 .
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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
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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)).