Trigonometric formulae

The basic trigonometric functions are sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot).

Trigonometric ratios in a right triangle are defined as follows:
Sinθ = Opposite side / Hypotenuse
Cosθ = Adjacent side / Hypotenuse
Tanθ = Opposite side / Adjacent side
Cscθ = 1 / Sinθ
Secθ = 1 / Cosθ
Cotθ = 1 / Tanθ

The Pythagorean trigonometric identity is: Sin²θ + Cos²θ = 1

To convert radians to degrees, multiply by (180/π). To convert degrees to radians, multiply by (π/180)

The sum and difference formulas are:
Sin(A ± B) = SinA * CosB ± CosA * SinB
Cos(A ± B) = CosA * CosB ∓ SinA * SinB

The double-angle formula for sine is: Sin(2θ) = 2 * Sinθ * Cosθ

The half-angle formulas are:
Sin(θ/2) = ± √[(1 – Cosθ) / 2]
Cos(θ/2) = ± √[(1 + Cosθ) / 2]
Tan(θ/2) = ± √[(1 – Cosθ) / (1 + Cosθ)]

Trigonometric values of special angles (0°, 30°, 45°, 60°, 90°) can be determined using reference triangles or the unit circle.

The law of sines states: (SinA / a) = (SinB / b) = (SinC / c), where A, B, C are angles and a, b, c are corresponding sides in a triangle. It’s used to solve triangles when given angle-side-angle or side-angle-side.

The law of cosines is: c² = a² + b² – 2ab * CosC. It’s used to solve triangles when given side-side-side (SSS), side-angle-side (SAS), or angle-side-angle (ASA).

The trigonometric identities are:
Tanθ = Sinθ / Cosθ
Cotθ = Cosθ / Sinθ

Trigonometric functions are used in fields like engineering, physics, astronomy, and navigation for calculating distances, angles, waves, and more.

Trigonometric functions are related to the unit circle by the coordinates (Cosθ, Sinθ) for any angle θ.

Trigonometric identities are proven using algebraic manipulations, substitution, and applying known trigonometric relationships.

Inverse trigonometric functions (Arcsin, Arccos, Arctan, Arccsc, Arcsec, Arccot) return angles corresponding to given ratios, with specific domains and ranges to ensure uniqueness of solutions.

The derivatives of trigonometric functions are used in calculus to find rates of change and slopes of trigonometric curves.

Trigonometric functions can be integrated using techniques like substitution, integration by parts, and trigonometric identities.

Triangles can be solved using trigonometric ratios such as sine, cosine, and tangent, based on the lengths of sides and measures of angles.

The general solution of a trigonometric equation includes all possible solutions within a specified interval, taking into account periodicity of trigonometric functions.