The branch of mathematics that studies relationships between the sides and angles of triangles. Trigonometry extends far beyond triangles, providing essential tools for modeling waves, circular motion, and periodic phenomena throughout science and engineering.
Trigonometry — from the Greek words trigonon (triangle) and metron (measure) — is the study of the relationships between the angles and sides of triangles. It originated in the ancient world as a practical tool for astronomy and land surveying, but has grown into a vast discipline that underlies much of modern mathematics, physics, and engineering.
At its core, trigonometry answers a simple question: if you know some measurements of a triangle, can you determine the rest? The answer is yes, and the six trigonometric functions — sine, cosine, tangent, cosecant, secant, and cotangent — are the tools that make it possible.
Beyond triangles, trigonometric functions turn out to be the perfect mathematical language for describing anything that repeats in a cycle: sound waves, light waves, alternating current, the motion of a pendulum, the orbit of a planet, and countless other periodic phenomena.
Trigonometry is the bridge between algebra and calculus. A solid understanding of trig functions and identities is essential for success in calculus, physics, and engineering courses.
Key areas where trigonometry is indispensable include:
Physics: Describing waves, oscillations, projectile motion, and forces
Engineering: Signal processing, structural analysis, electrical circuits
Astronomy: Measuring distances to stars, planetary orbits
Right Triangle Trigonometry
The foundation of trigonometry is the right triangle — a triangle containing one 90° angle. In a right triangle, the side opposite the right angle is called the hypotenuse (always the longest side), and the other two sides are called legs. Relative to a given acute angle θ, the legs are labeled as the opposite side (across from θ) and the adjacent side (next to θ).
SOH-CAH-TOA
The three primary trigonometric ratios are defined using the sides of a right triangle relative to an acute angle θ:
This triangle is half of an equilateral triangle. If the shortest side (opposite the 30° angle) has length 1, then the hypotenuse is 2 and the longer leg is √3.
Memorize the side ratios for the 45-45-90 and 30-60-90 triangles. They appear constantly in standardized tests, calculus, physics, and engineering problems. Knowing them by heart will save you enormous time.
The Unit Circle
The unit circle is a circle of radius 1 centered at the origin of a coordinate plane. It is one of the most powerful tools in trigonometry because it extends the definitions of sine and cosine beyond acute angles to all real numbers.
Definition on the Unit Circle
For any angle θ measured from the positive x-axis (counterclockwise is positive), the terminal side of the angle intersects the unit circle at a point (x, y). We define:
cos(θ) = x (the x-coordinate)
sin(θ) = y (the y-coordinate)
Since the radius is 1, this is consistent with SOH-CAH-TOA: cos(θ) = adjacent/hypotenuse = x/1 = x, and sin(θ) = opposite/hypotenuse = y/1 = y.
Key Angles on the Unit Circle
The following table lists the most important angles and their sine and cosine values:
0° (0 rad): cos = 1, sin = 0
30° (π/6): cos = √3/2, sin = 1/2
45° (π/4): cos = √2/2, sin = √2/2
60° (π/3): cos = 1/2, sin = √3/2
90° (π/2): cos = 0, sin = 1
120° (2π/3): cos = -1/2, sin = √3/2
135° (3π/4): cos = -√2/2, sin = √2/2
150° (5π/6): cos = -√3/2, sin = 1/2
180° (π): cos = -1, sin = 0
210° (7π/6): cos = -√3/2, sin = -1/2
225° (5π/4): cos = -√2/2, sin = -√2/2
240° (4π/3): cos = -1/2, sin = -√3/2
270° (3π/2): cos = 0, sin = -1
300° (5π/3): cos = 1/2, sin = -√3/2
315° (7π/4): cos = √2/2, sin = -√2/2
330° (11π/6): cos = √3/2, sin = -1/2
360° (2π): cos = 1, sin = 0
Reference Angles
A reference angle is the acute angle (between 0° and 90°) formed between the terminal side of an angle and the x-axis. Reference angles allow you to evaluate trig functions for any angle by relating them back to a first-quadrant angle.
Step 1: 240° is in Quadrant III (between 180° and 270°).
Step 2: Reference angle = 240° - 180° = 60°
So sin(240°) = -sin(60°) = -√3/2 and cos(240°) = -cos(60°) = -1/2 (both negative in Quadrant III).
Quadrant Signs — ASTC
The mnemonic "All Students Take Calculus" (ASTC) tells you which trig functions are positive in each quadrant:
Quadrant I (A — All): All trig functions are positive
Quadrant II (S — Sine): Only sine (and cosecant) are positive
Quadrant III (T — Tangent): Only tangent (and cotangent) are positive
Quadrant IV (C — Cosine): Only cosine (and secant) are positive
To evaluate any trig function at any angle: (1) find the reference angle, (2) evaluate the function at the reference angle, (3) apply the correct sign using ASTC. This three-step method works every time.
Radian Measure
Radians are the standard unit of angle measurement in mathematics. One radian is the angle subtended at the center of a circle by an arc equal in length to the radius.
When we treat the trig ratios as functions of a real variable, we unlock the full power of trigonometry. Each trigonometric function takes an angle as input and returns a ratio as output, and each has its own distinctive graph, domain, range, and period.
The sine function starts at the origin, rises to a maximum of 1 at π/2, returns to 0 at π, drops to a minimum of -1 at 3π/2, and returns to 0 at 2π, completing one full cycle.
|A| = amplitude (vertical stretch; the height from the midline to a peak)
2π/|B| = period (horizontal stretch; how long one cycle takes)
C/B = phase shift (horizontal translation)
D = vertical shift (moves the midline up or down)
Example: Graphing a Transformed Sine Function
Describe the graph of y = 3 sin(2x - π) + 1.
Amplitude: |A| = 3 (the graph oscillates 3 units above and below the midline)
Period: 2π/|B| = 2π/2 = π (one full cycle every π units)
Phase shift: C/B = π/2 (shifted π/2 units to the right)
Vertical shift: D = 1 (midline is y = 1 instead of y = 0)
The graph oscillates between y = 1 - 3 = -2 and y = 1 + 3 = 4.
The coefficient B affects the period, not the frequency. A larger B means a shorter period (more cycles packed into the same space). The frequency is f = |B|/(2π), representing how many cycles occur per unit.
Trigonometric Identities
Trigonometric identities are equations involving trig functions that are true for every value in their domain. They are essential for simplifying expressions, solving equations, and proving other mathematical results. Mastering these identities is critical for success in calculus.
Pythagorean Identities
These are derived directly from the Pythagorean theorem applied to the unit circle (x² + y² = 1):
sin A cos B = ½[sin(A + B) + sin(A - B)]
cos A cos B = ½[cos(A - B) + cos(A + B)]
sin A sin B = ½[cos(A - B) - cos(A + B)]
sin A + sin B = 2 sin[(A + B)/2] cos[(A - B)/2]
sin A - sin B = 2 cos[(A + B)/2] sin[(A - B)/2]
cos A + cos B = 2 cos[(A + B)/2] cos[(A - B)/2]
cos A - cos B = -2 sin[(A + B)/2] sin[(A - B)/2]
You don't need to memorize every identity at once. Start with the Pythagorean identities and sum/difference formulas — the rest can be derived from these. Understanding derivations is more valuable than rote memorization.
Inverse Trigonometric Functions
The inverse trigonometric functions answer the question: "What angle has this trig value?" Since trig functions are periodic and not one-to-one, we must restrict their domains to create proper inverse functions.
Arcsine: y = sin⁻¹(x) or y = arcsin(x)
Question it answers: "What angle between -π/2 and π/2 has a sine of x?"
Domain: [-1, 1]
Range: [-π/2, π/2] (i.e., -90° to 90°)
If sin(θ) = x, then θ = arcsin(x), where -π/2 ≤ θ ≤ π/2
Arccosine: y = cos⁻¹(x) or y = arccos(x)
Question it answers: "What angle between 0 and π has a cosine of x?"
Domain: [-1, 1]
Range: [0, π] (i.e., 0° to 180°)
If cos(θ) = x, then θ = arccos(x), where 0 ≤ θ ≤ π
Arctangent: y = tan⁻¹(x) or y = arctan(x)
Question it answers: "What angle between -π/2 and π/2 has a tangent of x?"
Domain: All real numbers (-∞, ∞)
Range: (-π/2, π/2) (i.e., -90° to 90°, exclusive)
If tan(θ) = x, then θ = arctan(x), where -π/2 < θ < π/2
Example: Evaluating Inverse Trig Functions
Find the exact value of each:
arcsin(1/2): What angle in [-π/2, π/2] has sin = 1/2? Answer: π/6 (30°)
arccos(-√2/2): What angle in [0, π] has cos = -√2/2? Answer: 3π/4 (135°)
arctan(1): What angle in (-π/2, π/2) has tan = 1? Answer: π/4 (45°)
arcsin(-1): What angle in [-π/2, π/2] has sin = -1? Answer: -π/2 (-90°)
Compositions of Trig and Inverse Trig Functions
When composing trig functions with their inverses, be careful about domain restrictions:
sin(arcsin(x)) = x for -1 ≤ x ≤ 1
arcsin(sin(x)) = x only for -π/2 ≤ x ≤ π/2
Example: Composition
Find cos(arcsin(3/5)).
Step 1: Let θ = arcsin(3/5). Then sin(θ) = 3/5 with θ in [-π/2, π/2].
Step 2: We need cos(θ). Using sin²(θ) + cos²(θ) = 1:
cos²(θ) = 1 - 9/25 = 16/25
Step 3: cos(θ) = 4/5 (positive because θ is in [-π/2, π/2])
A useful trick: when evaluating compositions like cos(arctan(x)), draw a right triangle. If arctan(x) = θ, then tan(θ) = x = x/1, so the opposite side is x and the adjacent side is 1. The hypotenuse is √(1 + x²), and you can read off any trig function directly.
Solving Trigonometric Equations
A trigonometric equation is an equation that involves trigonometric functions of unknown angles. Unlike identities (which are true for all values), trig equations are true only for specific values. Because trig functions are periodic, these equations typically have infinitely many solutions.
General Strategy
Isolate the trig function (get sin(x), cos(x), or tan(x) alone on one side).
Find the reference angle using inverse trig functions.
Identify all solutions in one period (usually [0, 2π) or [0°, 360°)).
Write the general solution by adding full periods: + 2nπ for sin/cos, or + nπ for tan (where n is any integer).
Example: Basic Trig Equation
Solve: 2 sin(x) - 1 = 0
Step 1: Isolate sin(x): sin(x) = 1/2
Step 2: Reference angle: arcsin(1/2) = π/6
Step 3: sin is positive in Quadrants I and II:
x = π/6 and x = π - π/6 = 5π/6
Step 4: General solution: x = π/6 + 2nπ or x = 5π/6 + 2nπ, where n is any integer.
Example: Equation Requiring Factoring
Solve: 2cos²(x) - cos(x) - 1 = 0 on [0, 2π)
Step 1: This is a quadratic in cos(x). Let u = cos(x): 2u² - u - 1 = 0
Step 2: Factor: (2u + 1)(u - 1) = 0
Step 3: u = -1/2 or u = 1, so cos(x) = -1/2 or cos(x) = 1
Step 1: Use the double-angle identity: 2sin(x)cos(x) = cos(x)
Step 2: Rearrange: 2sin(x)cos(x) - cos(x) = 0
Step 3: Factor: cos(x)[2sin(x) - 1] = 0
Step 4: cos(x) = 0: x = π/2, 3π/2
sin(x) = 1/2: x = π/6, 5π/6
Solutions: x = π/6, π/2, 5π/6, 3π/2
Example: Equation with Multiple Angles
Solve: tan(3x) = 1 on [0, 2π)
Step 1: Reference angle: arctan(1) = π/4
Step 2: General solution for 3x: 3x = π/4 + nπ
Step 3: Solve for x: x = π/12 + nπ/3
Step 4: Find values in [0, 2π) by substituting n = 0, 1, 2, 3, 4, 5:
x = π/12, 5π/12, 3π/4, 13π/12, 17π/12, 7π/4
Solutions: x = π/12, 5π/12, 3π/4, 13π/12, 17π/12, 7π/4
Common mistakes: (1) Dividing both sides by a trig function — this can cause you to lose solutions where that function equals zero. Always factor instead. (2) Forgetting that trig equations typically have multiple solutions in each period. Always check all quadrants.
Law of Sines and Law of Cosines
While SOH-CAH-TOA only works for right triangles, the Law of Sines and Law of Cosines allow us to solve any triangle — including oblique triangles (those without a right angle). These are arguably the two most powerful formulas in triangle trigonometry.
Law of Sines
In any triangle with sides a, b, c opposite angles A, B, C respectively:
a / sin(A) = b / sin(B) = c / sin(C)
Equivalently: sin(A)/a = sin(B)/b = sin(C)/c
Use the Law of Sines when you know:
AAS (Angle-Angle-Side): Two angles and a non-included side
ASA (Angle-Side-Angle): Two angles and the included side
SSA (Side-Side-Angle): Two sides and an angle opposite one of them (the ambiguous case)
Example: Using the Law of Sines (AAS)
In triangle ABC, A = 42°, B = 73°, and a = 15. Find side b.
Step 1: Find C: C = 180° - 42° - 73° = 65°
Step 2: Apply the Law of Sines: a/sin(A) = b/sin(B)
When given two sides and an angle opposite one of them, there may be zero, one, or two possible triangles. This is because the sine function gives the same value for supplementary angles (sin(θ) = sin(180° - θ)).
If the computed sine of the unknown angle is greater than 1: no triangle exists
If the sine equals 1: exactly one right triangle
If the sine is less than 1: check whether both the angle and its supplement create valid triangles (two solutions possible)
Example: Ambiguous Case
In triangle ABC, a = 8, b = 12, and A = 30°. How many triangles are possible?
When you know two sides and the included angle, the area is:
Area = ½ · a · b · sin(C)
When you know all three sides, use Heron's formula:
s = (a + b + c) / 2 (semi-perimeter)
Area = √[s(s - a)(s - b)(s - c)]
Example: Area with Two Sides and an Included Angle
Find the area of a triangle with sides a = 9, b = 14, and included angle C = 62°.
Area = ½ · 9 · 14 · sin(62°) = 63 · 0.8829 ≈ 55.62 square units
When deciding between the Law of Sines and Law of Cosines, ask: "Do I have a complete ratio (a side and its opposite angle)?" If yes, use the Law of Sines. If not, use the Law of Cosines. For SSS problems you must start with the Law of Cosines since you have no angles.
Polar Coordinates
Polar coordinates provide an alternative to the Cartesian (rectangular) coordinate system. Instead of locating a point by horizontal and vertical distances (x, y), polar coordinates use a distance from the origin (r) and an angle from the positive x-axis (θ).
Polar point: (r, θ)
r = distance from the origin (pole)
θ = angle from the positive x-axis (polar axis)
Converting Between Polar and Rectangular Coordinates
Polar coordinates are especially useful when working with curves that have rotational symmetry. Circles, spirals, and flower-like curves that would be complicated in Cartesian coordinates often have elegantly simple polar equations.
Applications of Trigonometry
Trigonometry is one of the most applied branches of mathematics. Its reach extends from ancient navigation to cutting-edge signal processing. Here are some of the most important real-world applications.
Navigation and Surveying
Trigonometry has been used for navigation since ancient times. By measuring angles to known landmarks or celestial objects, navigators can determine their position with remarkable accuracy.
Example: Angle of Elevation
From a point 200 meters from the base of a building, the angle of elevation to the top is 35°. How tall is the building?
Sine and cosine functions are the natural language of waves. Any wave — whether it's a sound wave, light wave, radio wave, or water wave — can be described using sinusoidal functions.
The general equation of a simple harmonic wave is:
y(t) = A sin(2πft + φ)
Where:
A = amplitude (maximum displacement)
f = frequency (cycles per second, measured in Hz)
t = time
φ = phase (initial offset)
Period T = 1/f (time for one complete cycle)
Example: Sound Wave
Middle C on a piano has a frequency of 261.63 Hz. Write the equation for this sound wave with amplitude 0.02 and no phase shift.
y(t) = 0.02 sin(2π · 261.63 · t)
y(t) = 0.02 sin(1643.8t)
The period is T = 1/261.63 ≈ 0.00382 seconds (about 3.82 milliseconds per cycle).
Simple Harmonic Motion
Objects on springs, pendulums, and many other physical systems exhibit simple harmonic motion (SHM), which is described by trigonometric functions:
Where ω = 2πf is the angular frequency (in radians per second).
Example: Mass on a Spring
A mass on a spring oscillates with amplitude 0.1 m and period 2 seconds. Find the position and velocity at t = 0.5 s if the mass starts at maximum displacement.
Trigonometry is essential for decomposing vectors into components and for computing resultant vectors — fundamental operations in physics and engineering.
A vector v with magnitude |v| at angle θ from the positive x-axis has components:
Step 4: Direction: θ = arctan(94.3/3.3) ≈ 88.0° from the positive x-axis
Electrical Engineering: AC Circuits
Alternating current (AC) voltage and current are sinusoidal, making trigonometry the essential mathematical language for electrical engineering:
V(t) = V₀ sin(2πft)
In the United States, household electricity uses V₀ ≈ 170 V and f = 60 Hz, giving the familiar "120 V AC" (the root-mean-square value is V₀/√2 ≈ 120 V).
Computer Graphics and Animation
Rotations in 2D computer graphics rely on trigonometry. To rotate a point (x, y) by angle θ around the origin:
x' = x cos(θ) - y sin(θ)
y' = x sin(θ) + y cos(θ)
This rotation matrix is the foundation of every 2D graphics engine, and its 3D extension (using Euler angles or quaternions) powers video games, CAD software, and CGI in films.
Example: Rotating a Point
Rotate the point (3, 4) by 90° counterclockwise around the origin.
x' = 3 cos(90°) - 4 sin(90°) = 3(0) - 4(1) = -4
y' = 3 sin(90°) + 4 cos(90°) = 3(1) + 4(0) = 3
New point: (-4, 3)
Trigonometry is truly everywhere. From the GPS receiver in your phone (which uses triangulation) to the Fourier transforms that compress your music files (which decompose signals into sine waves), trig functions are at work behind the scenes in virtually every piece of modern technology.
Hyperbolic Functions
The hyperbolic functions are analogues of the circular trigonometric functions, but defined using the hyperbola x² − y² = 1 instead of the circle x² + y² = 1. They appear naturally in areas like catenary curves, special relativity, and solutions to differential equations.
Catenary curve: A hanging chain or cable takes the shape y = a·cosh(x/a), not a parabola
Special relativity:Rapidity uses hyperbolic functions — velocities add via tanh, not simple addition
Integral results: Many integrals involving √(x² ± 1) are expressed using inverse hyperbolic functions
Example: Inverse Hyperbolic Functions
Since sinh is one-to-one, it has a true inverse:
sinh⁻¹(x) = ln(x + √(x² + 1))
This appears in calculus when evaluating:
∫ dx / √(x² + 1) = sinh⁻¹(x) + C = ln(x + √(x² + 1)) + C
Fourier Series
One of the most profound applications of trigonometry is the Fourier series: the idea that any periodic function can be decomposed into a sum of sines and cosines. This discovery by Joseph Fourier in 1807 revolutionized mathematics, physics, and engineering.
The Fourier Series Formula
For a periodic function f(x) with period 2L:
f(x) = a₀/2 + ∑ [aₙ cos(nπx/L) + bₙ sin(nπx/L)], n = 1 to ∞
where the Fourier coefficients are:
a₀ = (1/L) ∫ f(x) dx from −L to L
aₙ = (1/L) ∫ f(x) cos(nπx/L) dx
bₙ = (1/L) ∫ f(x) sin(nπx/L) dx
Example: Fourier Series of a Square Wave
The square wave f(x) = 1 for 0 < x < π, f(x) = −1 for −π < x < 0 has the Fourier series:
Only odd harmonics appear (by symmetry), and the coefficients decrease as 1/n. Adding more terms gives a better approximation — though overshoots near discontinuities persist (the Gibbs phenomenon).
Why Fourier Series Matter
Signal processing: Every audio or radio signal is decomposed into frequency components using Fourier analysis
Solving PDEs: The heat equation, wave equation, and Laplace's equation are solved by Fourier series (this was Fourier's original motivation)
Image compression: JPEG uses the Discrete Cosine Transform (DCT), a variant of Fourier analysis
Music: A musical note is a sum of harmonics — the Fourier spectrum determines the instrument's timbre
Quantum mechanics: Wavefunctions are expanded in Fourier series; the uncertainty principle is a direct consequence of Fourier analysis
From Fourier Series to Fourier Transform
The Fourier series works for periodic functions. For non-periodic functions, the sum becomes an integral — the Fourier Transform:
f̂(ω) = ∫ f(t) e⁻ⁱωᵗ dt (from −∞ to ∞)
This transforms a function of time into a function of frequency. The inverse transform recovers the original:
f(t) = (1/2π) ∫ f̂(ω) eⁱωᵗ dω (from −∞ to ∞)
The Fast Fourier Transform (FFT) algorithm, published by Cooley and Tukey in 1965, computes the discrete Fourier transform in O(n log n) time instead of O(n²). It has been called one of the most important algorithms of the 20th century — it powers everything from cell phone signals to MRI scanners to Shazam's music recognition.
Explore Trigonometry Lessons
Dive deeper into specific trigonometry topics with our focused lessons: