Trigonometry Calculator
Calculate trigonometric and inverse trigonometric functions. Optionally, compute the hypotenuse of a right triangle given two sides.
Results
About Trigonometric Functions: Trigonometry studies angles and the relationships between triangle sides. Functions like sine, cosine, tangent, and their inverses are used in math, physics, and engineering.
Practical Applications: Useful for students, engineers, architects, and anyone working with angles or right triangles.
Understanding SOH-CAH-TOA
The mnemonic SOH-CAH-TOA is the easiest way to remember the three primary trigonometric ratios for a right triangle. Each part of the mnemonic connects a trigonometric function to a ratio of two sides of the triangle relative to a given angle.
SOH: sin(angle) = Opposite / Hypotenuse
CAH: cos(angle) = Adjacent / Hypotenuse
TOA: tan(angle) = Opposite / Adjacent
The remaining three functions -- cosecant (csc), secant (sec), and cotangent (cot) -- are simply the reciprocals of sine, cosine, and tangent, respectively. For example, csc(angle) = 1 / sin(angle).
The Unit Circle and Radians vs. Degrees
The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane. It provides a visual framework for understanding all trigonometric functions, not just those in right triangles. Any point on the unit circle can be described as (cos(theta), sin(theta)), where theta is the angle measured from the positive x-axis.
Angles can be measured in degrees or radians. A full rotation is 360 degrees or 2pi radians. The conversion formulas are:
Radians = Degrees x (pi / 180)
Degrees = Radians x (180 / pi)
Radians are the standard unit in calculus and most scientific applications because they simplify many formulas. For instance, the derivative of sin(x) is cos(x) only when x is in radians.
Exact Values of Trig Functions — All Standard Angles
Complete reference table for all six trig functions at standard angles from 0° to 360°. These exact values appear on every trig exam — no calculator needed once you know the patterns.
| Degrees | Radians | sin | cos | tan | cot |
|---|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 | — |
| 30° | π/6 | 1/2 | √3/2 | 1/√3 | √3 |
| 45° | π/4 | √2/2 | √2/2 | 1 | 1 |
| 60° | π/3 | √3/2 | 1/2 | √3 | 1/√3 |
| 90° | π/2 | 1 | 0 | — | 0 |
| 120° | 2π/3 | √3/2 | −1/2 | −√3 | −1/√3 |
| 135° | 3π/4 | √2/2 | −√2/2 | −1 | −1 |
| 150° | 5π/6 | 1/2 | −√3/2 | −1/√3 | −√3 |
| 180° | π | 0 | −1 | 0 | — |
| 210° | 7π/6 | −1/2 | −√3/2 | 1/√3 | √3 |
| 225° | 5π/4 | −√2/2 | −√2/2 | 1 | 1 |
| 240° | 4π/3 | −√3/2 | −1/2 | √3 | 1/√3 |
| 270° | 3π/2 | −1 | 0 | — | 0 |
| 300° | 5π/3 | −√3/2 | 1/2 | −√3 | −1/√3 |
| 315° | 7π/4 | −√2/2 | √2/2 | −1 | −1 |
| 330° | 11π/6 | −1/2 | √3/2 | −1/√3 | −√3 |
| 360° | 2π | 0 | 1 | 0 | — |
— = undefined. Colour bands: white = Q1 (0°–90°), blue = Q2, yellow = Q3, green = Q4. In Q1 all functions are positive. In Q2 only sin is positive. In Q3 only tan/cot are positive. In Q4 only cos is positive.
Inverse Trigonometric Functions and Applications
Inverse trigonometric functions (arcsin, arccos, arctan) work in the opposite direction of their standard counterparts. Instead of taking an angle and returning a ratio, they take a ratio and return an angle. For instance, arcsin(0.5) = 30 degrees because sin(30 degrees) = 0.5.
Keep in mind that inverse trig functions have restricted ranges to ensure they return a single value: arcsin and arctan return values between -90 and 90 degrees, while arccos returns values between 0 and 180 degrees.
Trigonometry has countless real-world applications. Engineers use it to calculate forces on structures and design bridges. Navigators rely on it for determining positions and plotting courses. In physics, trigonometric functions describe wave motion, oscillations, and alternating current circuits. Computer graphics use trig functions to rotate and transform objects on screen, and surveyors use angles and trigonometric ratios to measure distances and elevations accurately.
Essential Trigonometric Identities
Trigonometric identities are equations that hold true for all values of the variable. They are used to simplify expressions, solve equations, and derive new results. Mastering these identities is essential for calculus, physics, and engineering.
Pythagorean Identities
Double-Angle Formulas
Angle Addition Formulas
Law of Sines and Law of Cosines
SOH-CAH-TOA only works in right triangles. For any triangle — whether acute, obtuse, or right — the Law of Sines and Law of Cosines provide powerful tools to find unknown sides and angles.
Law of Sines
Use this when you know two angles and one side (AAS or ASA), or two sides and a non-included angle (SSA). It states that each side of a triangle is proportional to the sine of its opposite angle.
Law of Cosines
Use this when you know all three sides (SSS) or two sides and the included angle (SAS). It generalizes the Pythagorean theorem — when C = 90°, cos(90°) = 0, and it reduces to a² + b² = c².
Practical example: A surveyor stands at point C, 500 m from landmark A and 700 m from landmark B, with an angle of 62° at C. The distance AB = √(500² + 700² − 2 × 500 × 700 × cos(62°)) ≈ 638 m. This is a direct application of the Law of Cosines in land surveying.
Step-by-Step Examples
Example 1: Find the height of a building
A person stands 30 meters from the base of a building and measures the angle of elevation to the top as 58°. How tall is the building?
tan(58°) = opposite / adjacent = height / 30
height = 30 × tan(58°)
height = 30 × 1.6003 ≈ 48.0 meters
Example 2: Find an angle in a right triangle
A ramp rises 1.2 meters over a horizontal run of 5 meters. What angle does the ramp make with the ground?
tan(θ) = opposite / adjacent = 1.2 / 5 = 0.24
θ = arctan(0.24)
θ ≈ 13.5°
Example 3: Degrees ↔ Radians conversion
Convert 135° to radians, and convert 5π/6 radians to degrees.
135° × (π / 180) = 3π/4 ≈ 2.356 radians
(5π/6) × (180 / π) = 900/6 = 150°
Most Searched Trig Values — Quick Answers
Exact answers to the most common trigonometry lookups. No calculator needed once you know these.
| Expression | Exact Value | Decimal | Note |
|---|---|---|---|
| sin 30° | 1/2 | 0.5 | 30-60-90 triangle |
| sin 45° | √2/2 | 0.7071 | 45-45-90 triangle |
| sin 60° | √3/2 | 0.8660 | 30-60-90 triangle |
| sin 90° | 1 | 1 | Maximum value of sin |
| sin 0° | 0 | 0 | Minimum absolute value |
| cos 30° | √3/2 | 0.8660 | Same as sin 60° |
| cos 45° | √2/2 | 0.7071 | Same as sin 45° |
| cos 60° | 1/2 | 0.5 | Same as sin 30° |
| cos 90° | 0 | 0 | cos is zero at 90° |
| cos 0° | 1 | 1 | Maximum value of cos |
| tan 30° | 1/√3 | 0.5774 | ≈ 0.577350… |
| tan 45° | 1 | 1 | Opposite = Adjacent |
| tan 60° | √3 | 1.7321 | ≈ 1.732050… |
| tan 90° | undefined | — | Division by zero |
| arcsin(0.5) | 30° | π/6 | sin⁻¹(0.5) |
| arccos(0.5) | 60° | π/3 | cos⁻¹(0.5) |
| arctan(1) | 45° | π/4 | tan⁻¹(1) |
| arctan(√3) | 60° | π/3 | tan⁻¹(1.732) |