Home / Trigonometry / Identities & Equations

Identities & Equations

The toolkit for simplifying and solving — from Pythagorean identities to double-angle formulas.

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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.

All these identities are on the formula sheet. But proving them yourself — using the unit circle and geometric arguments — builds deep understanding.

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

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

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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

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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

Related Topics

The Unit CircleWhere all identities begin — the geometric foundation. IntegralsTrig substitution and power-reduction use identities constantly. EquationsSolving trig equations uses exactlythe same algebraic techniques. Formula SheetQuick-reference for all trigonometric identities and formulas.