In This Lesson Fundamental Identities Sum & Difference Formulas Double & Half-Angle Formulas uscW/yLLPjz+19NdXVIyIyoumRR8jRNIfKasMlHta/pk5dtkpOto9GcTWtQcoykKRFsJlGztbf70JsVrOWnXhnadDWdAnywvlBVimel/ZXmBRgzHZEW24CnuByqtd4VxEXxkwgnIZAnqtnv95cFzrddvuLIq8hupZOaYgBAZZZMkXrdA2TsA83eKHZlNvBxsrdt9hWZdkVWH+Ryzg2VO/V0oYIA9q9sKfV/e4P+7ESuls5HQQV2MQhaiGK1D//NPIlTg/LNWzi+F6AinaYDzR3O5beK1PzPhs/iwOwE2CJjRGrxHdW65R+r21BSNEvJnuSjsWkZipmxMmyC6bJE8Zo0BwUCiEicLIqNAu7LRUOSS4nOIKp/mgj9Y9LIsimJF/DIGBQ9opvh1dY+xRnx6Gx0HyB2+0Z0J/huGrbLVzhjVM72tcFfHbaxqSTlFMGRAFdA5NvZERvnKuARopnTBhEkd91HOGuy6MzGLYvy9xAAHB2A87Zv8LPSekj/N6IwY7Jz6wj9g87jFrJ/VheXZSqU8anBV0JsIgtze6663GpKR1iaLvpXNH+LrILEzYZ1AQnY4FerPyfGeYhYKkaBNrWp0DCi9OiKanyIIuKJ6mBenfGTyDmxwM3Q/jtruBDeCSPRWlnM+QJX9knb1XfV2c7mocNflUxX7PprmfteROuMGbyFq1QHPEhUbq0JPIF9Ba7EsEcm7eO Solving Trig Equations Inverse Trig Functions 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 Fundamental Identities Pythagorean: Reciprocal: csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θQuotient: 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
These let you expand trig functions of sums — crucial for Fourier analysis , deriving trig derivatives , and signal processing .
Double & Half-Angle Formulas Double angle: Half angle:
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:
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sin⁻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)).