In This Lesson Fundamental Identities 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 Sum & Difference Formulas 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 Double & Half-Angle Formulas 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 .
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
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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)).