Trigonometry Calculator — Compute Sin, Cos, Tan & Inverse Trig

The Formula

Trigonometry is the branch of mathematics that studies relationships between angles and sides of triangles. The six trigonometric functions and their inverses allow you to convert between an angle and its trigonometric ratio. Sine, cosine, and tangent are the primary functions; cosecant, secant, and cotangent are their reciprocals. The inverse functions (arcsin, arccos, arctan) return the principal angle whose sine, cosine, or tangent equals a given value. These relationships are fundamental to geometry, physics, engineering, navigation, and wave analysis, forming the backbone of periodic and oscillatory phenomena modeling.

Variable Definitions

How to Use This Calculator

1. 1

   Select the mode: "Angle → Trig Values" to compute all six trigonometric values from an angle, or "Trig Value → Angle" to find the principal angle from a trigonometric value.

2. 2

   In Angle mode, enter any angle in degrees (e.g., 30, 45, 60, 90) and instantly see sin, cos, tan, csc, sec, and cot values with appropriate precision.

3. 3

   In Trig Value mode, choose the trigonometric function (sin, cos, tan, csc, sec, or cot) and enter its numeric value. The calculator returns the principal angle and the reference angle.

4. 4

   Pay attention to domain restrictions: sin and cos accept values only in [-1, 1], while tan accepts any real number. For csc and sec, the absolute value must be ≥ 1.

5. 5

   Use the principal angle for the direct inverse result and the reference angle for understanding the acute-angle relationship in right triangle contexts.

Quick Reference

Common Applications

• Right triangle solving: given one angle and one side, find all missing sides and angles using SOH CAH TOA, essential for construction, surveying, and navigation
• Physics and engineering: analyzing periodic motion such as pendulums, springs, waves (sound, light, water), and alternating current (AC) circuits where sinusoidal functions describe oscillation
• Computer graphics and game development: rotating objects, calculating distances and angles in 2D/3D space, and implementing camera movements using trigonometric transformations
• Navigation and GPS: converting between bearing angles and coordinate displacements, calculating great-circle distances on Earth's surface, and maritime/aviation heading corrections
• Architecture and structural engineering: calculating roof pitches, ramp slopes, load angles, and the forces acting on structural members at various angles of incidence

The six trigonometric functions are defined as ratios of right triangle sides. SOH CAH TOA is a mnemonic for the three primary functions. The reciprocal functions are the multiplicative inverses.

Understanding the Concept

Trigonometry is a fundamental branch of mathematics that deals with the relationships between angles and sides of triangles, particularly right triangles. The word comes from Greek "trigonon" (triangle) and "metron" (measure). At its core, trigonometry provides six functions that connect an angle to the ratios of triangle sides. The three primary functions are sine (sin), cosine (cos), and tangent (tan), commonly remembered by the mnemonic SOH CAH TOA: Sine = Opposite over Hypotenuse, Cosine = Adjacent over Hypotenuse, Tangent = Opposite over Adjacent. The three reciprocal functions are cosecant (csc = 1/sin), secant (sec = 1/cos), and cotangent (cot = 1/tan). These functions have specific domains and ranges: sine and cosine accept any real angle and return values between -1 and 1; tangent accepts any angle except odd multiples of 90° (where cos = 0) and returns all real numbers. The reciprocal functions have complementary domains: csc and sec are undefined for angles where sin or cos equal zero, and their absolute values are always greater than or equal to 1. The inverse trigonometric functions (arcsin, arccos, arctan) reverse this process, returning the principal angle for a given trigonometric value. PrincipalThe original sum of money borrowed or invested, excluding any interest or earnings. angles have restricted ranges: arcsin returns angles in [-90°, 90°], arccos in [0°, 180°], and arctan in (-90°, 90°). For example, sin(30°) = 1/2, so arcsin(1/2) = 30°, and the reference angle (the acute angle to the x-axis) is also 30°. For negative values, arcsin(-1/2) = -30°, but the reference angle (absolute value) remains 30°. Trigonometric functions are periodic: sine and cosine have periods of 360° (2π radians), while tangent has a period of 180° (π radians). This periodicity means there are infinitely many angles producing the same trigonometric value. The principal angle is the canonical representative, typically the one closest to 0°. Understanding these functions is essential for analyzing any phenomenon involving rotation, oscillation, waves, or periodic behavior, from the orbit of planets to the vibration of guitar strings to the analysis of alternating current in electrical engineering. Trigonometry also underpins the Fourier transform, a mathematical tool that decomposes complex signals into their constituent sinusoidal frequencies, which is fundamental to digital signal processing, image compression (JPEG), and audio analysis.

Frequently Asked Questions

Sources & References