The six main trigonometric functions are: Sine (sin), Cosine (cos), Tangent (tan), Cosecant (csc), Secant (sec), and Cotangent (cot). The first three are primary functions, while cosecant, secant, and cotangent are their reciprocals. For a right triangle with angle θ: sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse, tan(θ) = opposite/adjacent, csc(θ) = hypotenuse/opposite, sec(θ) = hypotenuse/adjacent, and cot(θ) = adjacent/opposite.

To convert degrees to radians, multiply by π/180 (approximately 0.01745). To convert radians to degrees, multiply by 180/π (approximately 57.296). For example, 90° = 90 × π/180 = π/2 radians. Common conversions: 30° = π/6, 45° = π/4, 60° = π/3, 180° = π, 360° = 2π.

Sin(30°) equals 0.5 or 1/2. This is one of the special angles in trigonometry. In radians, this is sin(π/6) = 1/2. This value comes from the properties of a 30-60-90 right triangle, where the side opposite the 30° angle is half the length of the hypotenuse.

Inverse trig functions (arcsin, arccos, arctan) find the angle given a ratio. For example, if sin(θ) = 0.5, then θ = arcsin(0.5) = 30° or π/6 radians. On calculators, these are often labeled as sin⁻¹, cos⁻¹, and tan⁻¹. Note that inverse functions have restricted ranges: arcsin returns values in [-90°, 90°], arccos in [0°, 180°], and arctan in (-90°, 90°).

The Pythagorean identity states that sin²(θ) + cos²(θ) = 1 for any angle θ. This is derived from the Pythagorean theorem applied to a right triangle on the unit circle. Two other related identities are: 1 + tan²(θ) = sec²(θ) and 1 + cot²(θ) = csc²(θ). These identities are fundamental in simplifying trigonometric expressions.

Hyperbolic functions are analogues of trigonometric functions using hyperbolas instead of circles. The main ones are sinh (hyperbolic sine), cosh (hyperbolic cosine), and tanh (hyperbolic tangent). They’re defined using exponential functions: sinh(x) = (eˣ – e⁻ˣ)/2, cosh(x) = (eˣ + e⁻ˣ)/2, and tanh(x) = sinh(x)/cosh(x). They appear in physics, engineering, and calculus.

Double angle formulas express trig functions of 2θ in terms of θ: sin(2θ) = 2sin(θ)cos(θ), cos(2θ) = cos²(θ) – sin²(θ) = 2cos²(θ) – 1 = 1 – 2sin²(θ), and tan(2θ) = 2tan(θ)/(1 – tan²(θ)). These formulas are useful for simplifying expressions, solving equations, and in calculus for integration.

Simply enter the negative angle value. The calculator handles negative angles correctly using the properties of trig functions: sin(-θ) = -sin(θ) (odd function), cos(-θ) = cos(θ) (even function), and tan(-θ) = -tan(θ) (odd function). For example, sin(-30°) = -0.5, cos(-30°) = 0.866, and tan(-30°) = -0.577.