Identities & Equations

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

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)).

← Previous LessonThe Unit Circle Next Lesson →Applications