Trigonometry

The double-angle formulas express sin(2θ), cos(2θ), and tan(2θ) in terms of sin θ, cos θ, and tan θ. They arise as the special case A = B of the sum formulas and appear throughout calculus (integrating powers of sine and cosine), physics (wave intensity, which depends on amplitude squared), and engineering (power calculations in AC circuits). Cosine's double-angle formula has three equivalent forms, related by the Pythagorean identity, each useful in different contexts.

The half-angle formulas express sin(θ/2), cos(θ/2), and tan(θ/2) in terms of cos θ. They are obtained by solving the double-angle cosine formula for sin²θ and cos²θ and then substituting θ/2 for θ, which is why they involve a square root and require choosing the correct sign based on which quadrant θ/2 falls in. They're used to find exact values at angles like 15° or 22.5° and to simplify integrals involving trigonometric functions via the Weierstrass (tangent half-angle) substitution.

The sum and difference formulas express the sine, cosine, and tangent of a sum or difference of two angles in terms of the sines, cosines, and tangents of the individual angles. They let you compute exact trigonometric values for angles like 15° or 75° that are not standard angles themselves but can be written as sums or differences of standard angles such as 30°, 45°, and 60°. These formulas are foundational: nearly every other trigonometric identity — double-angle, half-angle, product-to-sum — can be derived from them.

A trigonometric equation is an equation involving one or more trigonometric functions of an unknown angle, such as 2sinx − 1 = 0. Because trigonometric functions are periodic, such equations typically have infinitely many solutions, which are usually reported either as a full set differing by multiples of the period or restricted to one period such as [0°, 360°). Solving them combines inverse trigonometric functions (to find a first solution), reference angles and symmetry (to find all solutions in a period), and algebraic techniques like factoring or identity substitution when the equation involves more than one trig function.

The sum-to-product formulas rewrite a sum or difference of two sines (or two cosines) as a product of sines and cosines of half-sum and half-difference angles. They are the algebraic inverse of the product-to-sum formulas, and they are obtained directly from the angle sum and difference identities by adding or subtracting pairs of equations and substituting new variables for the half-sum and half-difference. These identities are indispensable whenever a sum of two oscillations needs to be seen as a single oscillation — the classic example is acoustic beats, where two tones of nearly equal frequency combine into one tone whose amplitude itself oscillates slowly.

Trigonometric substitution is a technique for simplifying algebraic expressions containing √(a²−x²), √(a²+x²), or √(x²−a²) by replacing x with a trigonometric function of a new variable θ. Each radical form is matched to the substitution that makes a Pythagorean identity collapse the expression under the square root into a single trig function, removing the radical entirely. The technique is the bridge between the Pythagorean identities (sin²θ+cos²θ=1, 1+tan²θ=sec²θ) and integral calculus, and it is the standard method for evaluating integrals whose integrand contains these three radical forms, as well as for simplifying geometric expressions such as chord lengths and areas that naturally involve a difference or sum of squares.

The inverse trigonometric functions — arcsin, arccos, and arctan (also written sin⁻¹, cos⁻¹, tan⁻¹) — undo the trigonometric functions, returning the angle whose sine, cosine, or tangent is a given value. Because sine, cosine, and tangent are periodic and not one-to-one over their full domains, each inverse must be restricted to a specific range to be well-defined: arcsin has range [−90°, 90°], arccos has range [0°, 180°], and arctan has range (−90°, 90°). These functions are essential for solving trigonometric equations, finding angles in applied problems (navigation, physics, engineering), and appear in calculus as antiderivatives of certain rational and algebraic functions.

The law of tangents relates the sum and difference of two sides of a triangle to the tangent of the half-sum and half-difference of their opposite angles. Historically it was valued (before calculators) because it let mathematicians solve the ambiguous SAS case of a triangle — two sides and the included angle — using only tangent tables, avoiding the awkward square roots of the law of cosines. Today it's mostly a curiosity and a good exercise in the sum-and-difference identities, since the law of cosines and law of sines handle triangle-solving more directly with modern computation.

The graphs of sine, cosine, and tangent are periodic waveforms whose shape is controlled by four parameters in the general form y = A sin(Bx + C) + D: the amplitude A (half the vertical distance between max and min), the period 2π/|B| (horizontal length of one full cycle), the phase shift −C/B (horizontal translation), and the vertical shift D (midline height). Tangent, cotangent, secant, and cosecant have their own characteristic shapes, including vertical asymptotes where the underlying function is undefined. Understanding how each parameter transforms the base graph is essential for modeling periodic phenomena like tides, sound waves, and seasonal temperature cycles.

The unit circle is a circle of radius 1 centered at the origin. It provides the modern definition of sine and cosine for ANY angle — not just angles inside a right triangle — by relating an angle to the coordinates of the point it sweeps out on the circle. This single idea extends trigonometry from acute angles in triangles to all real numbers, including negative angles and angles greater than 360°.

The trigonometric functions — sine, cosine, tangent, and their reciprocals cosecant, secant, cotangent — relate angles to ratios of sides in a right triangle, and more generally to coordinates on the unit circle. They are the essential tool for describing anything periodic: waves, oscillations, rotations, and circular motion.

Trigonometric identities are equations relating trig functions that hold for every value of the angle involved. They let you rewrite one trigonometric expression as another — simplifying integrals, solving equations, and proving other identities — much as algebraic identities like (a+b)² = a²+2ab+b² let you rewrite polynomial expressions.

The law of sines relates the lengths of the sides of any triangle to the sines of its opposite angles. For a triangle with sides a, b, c opposite angles α, β, γ, the ratio of each side to the sine of its opposite angle is constant and equals the diameter of the triangle's circumcircle. The law lets you solve oblique (non-right) triangles by triangulation: given two angles and a side (AAS/ASA), or two sides and a non-included angle (SSA), you can find the remaining parts. The SSA case is the notorious 'ambiguous case', where the given data can correspond to zero, one, or two valid triangles.

The law of cosines generalizes the Pythagorean theorem to any triangle, not just right triangles. For a triangle with sides a, b, c opposite angles α, β, γ, it expresses the square of one side in terms of the other two sides and the cosine of their included angle. When the included angle is 90°, cos(90°) = 0 and the formula reduces exactly to the Pythagorean theorem. The law of cosines is used to solve triangles given SAS (two sides and the included angle) or SSS (three sides), cases the law of sines cannot handle directly.

The angle of elevation is the angle between the horizontal and an observer's line of sight to an object above the horizontal; the angle of depression is the analogous angle to an object below the horizontal, measured from the horizontal down to the line of sight. These two angles are equal whenever the same horizontal line is shared by two observers looking at each other (a consequence of alternate interior angles for a line crossing two parallels), which is why a person on a cliff sighting a boat at a given angle of depression corresponds exactly to the boat's crew sighting the person at the same angle of elevation. Both angles turn a real-world sighting problem into a right triangle, letting a single measured angle and one known distance (height or horizontal range) determine every other length via the tangent, sine, or cosine ratio — the core technique behind surveying, navigation, and indirect height measurement.

Every complex number z = a + bi can be located as a point in the complex plane and described not just by its coordinates (a, b) but by its distance from the origin (the modulus r) and the angle its position vector makes with the positive real axis (the argument θ). Rewriting z in terms of r and θ instead of a and b gives the trigonometric (polar) form z = r(cosθ + i sinθ), which turns multiplication and division of complex numbers — normally messy FOIL-and-simplify arithmetic — into simple rules of adding or subtracting angles and multiplying or dividing moduli. This is the bridge between complex-number algebra and trigonometry, and it is the form that makes De Moivre's theorem and roots of complex numbers tractable.

De Moivre's theorem, named after Abraham de Moivre (1667–1754), states that raising a complex number in trigonometric form to an integer power n simply multiplies its argument by n and raises its modulus to the n-th power. This turns an otherwise painful repeated multiplication of complex numbers into a one-line computation, and — run in reverse with fractional exponents — it also produces a clean formula for the n distinct n-th roots of any complex number, evenly spaced around a circle in the complex plane. It is one of the most-used shortcuts in complex analysis and underlies applications from AC circuit phasors to signal processing (roots of unity in the discrete Fourier transform).

Trigonometry is the language that connects a vector's magnitude-and-direction description to its component description. Given a vector's length r and the angle θ it makes with the positive x-axis, its horizontal and vertical components are r cosθ and r sinθ — exactly the coordinates of a point on a circle of radius r, the same construction that defines sine and cosine on the unit circle. Conversely, given components, the magnitude comes from the Pythagorean theorem and the direction from the arctangent. This two-way translation is what makes vectors practical for physics and engineering: forces, velocities, and displacements are naturally described by magnitude and direction, but must be broken into components (or recombined) to add, and trigonometry is the tool that does the breaking and recombining.

Spherical trigonometry studies triangles drawn on the surface of a sphere, whose sides are arcs of great circles (the shortest paths between two points on a sphere) rather than straight line segments. Because the surface is curved, the familiar plane rules — angles summing to 180°, similar triangles, the ordinary law of sines and law of cosines — no longer hold exactly; spherical triangles have angle sums strictly greater than 180°, and their own analogues of the sine and cosine rules relate sides (measured as angles subtended at the sphere's center) to the triangle's vertex angles. This is not an abstract curiosity: it is the mathematics behind navigation, astronomy, and geodesy, since the Earth is (to a good approximation) a sphere and the shortest flight or ship route between two distant cities follows a great-circle arc.

The standard area formula for a triangle, (1/2)·base·height, requires knowing an altitude, which is often inconvenient to measure directly. The trigonometric area formula replaces the height with a side length and the sine of the included angle, so that the area of any triangle can be computed from two sides and the angle between them (SAS data) without ever constructing an altitude. This is exactly the same formula that underlies the cross-product magnitude formula for the area of a parallelogram in vector geometry, and it combines naturally with the Law of Sines and Law of Cosines to solve for areas of triangles specified in any of the standard ways (SSS, SAS, ASA).

Simple harmonic motion (SHM) is the repetitive back-and-forth motion of a system — a mass on a spring, a pendulum swinging through small angles, a vibrating guitar string — whose displacement over time is exactly a sinusoidal function. The position is modeled as x(t) = A cos(ωt + φ), where A is the amplitude (maximum displacement), ω is the angular frequency (radians per unit time), and φ is the phase constant that fixes where in the cycle the motion starts. This model arises directly from Newton's second law applied to a restoring force proportional to displacement (F = −kx), and the same trigonometric machinery used to analyze the graphs of sine and cosine — period, amplitude, phase shift — describes velocity, acceleration, energy, and frequency of the oscillating system.

Every point in the plane can be located either by rectangular coordinates (x, y) measured along perpendicular axes, or by polar coordinates (r, θ) giving a distance from the origin and an angle from the positive x-axis. Because a polar point sits at the vertex of a right triangle formed by dropping a perpendicular to the x-axis, the unit-circle definitions of sine and cosine translate directly between the two systems: x = r cosθ and y = r sinθ convert polar to rectangular, while r = √(x²+y²) and tanθ = y/x (with quadrant care) convert rectangular to polar. These conversions are essential for working with polar equations, complex numbers in polar form, and any physical situation — orbits, radar, robotics — where distance-and-angle data must be combined with Cartesian data.