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Trigonometry Calculator

Calculate sine, cosine, tangent, inverse trigonometric functions, solve right triangles, and explore unit circle values instantly.

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Trig Functions Calculator

Results

sin(θ)
0.5
cos(θ)
0.866
tan(θ)
0.577
csc(θ)
2
sec(θ)
1.155
cot(θ)
1.732

Right Triangle Solver

Results

Hypotenuse
5
Angle α
36.87°
Angle β
53.13°
Area
6
Perimeter
12

Inverse Trigonometry

Results

Angle (Degrees)
30°
Angle (Radians)
0.5236

Unit Circle Calculator

Results

Quadrant
II
Radians
2.094
Reference Angle
60°
sin
0.866
cos
-0.5
tan
-1.732

SOHCAHTOA Calculator

Fill any two values to solve. Angle in degrees.

Results

Solution
Enter values to compute
Opposite
—
Adjacent
—
Hypotenuse
—
Angle
—

Trigonometric Values Reference Table

Memorize these key values to solve problems faster.

Anglesincostan
0°010
30°0.50.8660.577
45°0.7070.7071
60°0.8660.51.732
90°10Undefined
180°0-10
270°-10Undefined
360°010

What Is Trigonometry?

Trigonometry is the branch of mathematics that studies relationships between the sides and angles of triangles.

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Definition

Trigonometry is derived from Greek words meaning "triangle measure." It defines relationships between angles and side lengths, especially in right-angled triangles.

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Angles & Sides

Every angle in a triangle corresponds to a unique ratio of side lengths. The six trigonometric functions—sin, cos, tan, csc, sec, cot—express those ratios.

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Right Triangles

A right triangle has a 90° angle. Its sides are labeled opposite, adjacent, and hypotenuse relative to a reference angle, forming the basis of SOHCAHTOA.

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Real-World Use

From bridge design to GPS, video games to astronomy, trigonometry powers calculations that involve distance, height, angles, and periodic motion.

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Beyond Triangles

Trig functions extend to the unit circle, describing periodic waves used in sound, light, electricity, signal processing, and quantum physics.

π

Why It Matters

Trigonometry bridges algebra and geometry, providing the mathematical language for engineering, science, architecture, and modern computing.

How to Use the Trigonometry Calculator

Five simple steps to instant results.

1

Choose Calculator Type

Pick a tab: trig functions, right triangle solver, inverse trig, unit circle, or SOHCAHTOA.

2

Enter Known Values

Type your angle, side lengths, or ratio into the input fields.

3

Select Degrees or Radians

Switch the unit selector to match your problem.

4

Calculate Results

Click Calculate to instantly view all related trig outputs.

5

Interpret Outputs

Use the reference tables and formulas below to verify and learn from the results.

Trigonometric Formulas

Memorize SOHCAHTOA and the Pythagorean identities to handle any trig problem.

Sine

sin θ = Opposite / Hypotenuse

Where Opposite is the side across from angle θ and Hypotenuse is the longest side of a right triangle.

Cosine

cos θ = Adjacent / Hypotenuse

Adjacent is the side next to angle θ (not the hypotenuse).

Tangent

tan θ = Opposite / Adjacent

Tangent equals sin θ ÷ cos θ and represents slope in many real-world applications.

Reciprocals

csc θ = 1/sin θ   sec θ = 1/cos θ   cot θ = 1/tan θ

The three reciprocal trig functions: cosecant, secant, and cotangent.

Pythagorean Identity

sin²θ + cos²θ = 1

True for any angle θ. Derived from the Pythagorean theorem on the unit circle.

Pythagorean Theorem

a² + b² = c²

In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.

The Unit Circle Explained

The unit circle is the foundation of modern trigonometry.

Definition

The unit circle is a circle of radius 1 centered at the origin of the coordinate plane. For any angle θ measured from the positive x-axis, the point (cos θ, sin θ) lies on the circle.

Angle Measurements

Angles can be measured in degrees (0° to 360°) or radians (0 to 2π). A full rotation equals 360° or 2π radians. To convert: radians = degrees × π/180.

Quadrants

The plane divides into four quadrants. Quadrant I (0–90°): all positive. II (90–180°): sin positive. III (180–270°): tan positive. IV (270–360°): cos positive. Remember "All Students Take Calculus."

Reference Angles

A reference angle is the acute angle between the terminal side of θ and the x-axis. It lets you compute trig values for any angle using values you already know from Quadrant I.

Right Triangle Worked Examples

See the calculator in action with step-by-step solutions.

Example 1: 3-4-5 Triangle

Given: Side A = 3, Side B = 4

Hypotenuse: c = √(3² + 4²) = √25 = 5

Angle α: arctan(3/4) ≈ 36.87°

Angle β: 90° − 36.87° = 53.13°

Area: (1/2) × 3 × 4 = 6

Perimeter: 3 + 4 + 5 = 12

Example 2: 45° Angle, Hypotenuse = 10

Given: Angle = 45°, Hypotenuse = 10

Opposite: 10 × sin(45°) ≈ 7.07

Adjacent: 10 × cos(45°) ≈ 7.07

Other Angle: 90° − 45° = 45°

Area: (1/2) × 7.07 × 7.07 ≈ 25

Perimeter: 7.07 + 7.07 + 10 ≈ 24.14

Real-World Applications of Trigonometry

Trig is everywhere — here are some industries that rely on it.

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Engineering

Designing gears, calculating stress angles, and modeling vibrations.

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Architecture

Calculating roof slopes, structural loads, and building proportions.

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Surveying

Measuring distances and elevations using triangulation.

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Physics

Modeling waves, projectile motion, and electromagnetic fields.

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Navigation

GPS, aviation routes, and maritime course plotting.

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Computer Graphics

Rotations, transformations, and 3D rendering in games and CGI.

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Construction

Cutting angles, framing, and ensuring structural alignment.

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Audio & Signal

Sine waves model sound, music synthesis, and signal processing.

Why Use a Trigonometry Calculator?

⚡

Fast Calculations

Get instant answers for trig functions, triangle sides, and angles without manual computation.

✓

Better Accuracy

Eliminate rounding errors and arithmetic mistakes common in hand calculations.

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Educational Support

Review formulas, work through examples, and reinforce classroom learning.

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Triangle Solving

Find missing sides and angles with one click using SOHCAHTOA.

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Angle Conversions

Switch seamlessly between degrees and radians for any problem.

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Mobile Friendly

Solve problems on your phone, tablet, or laptop — anywhere, anytime.

Frequently Asked Questions

Trigonometry is a branch of mathematics that studies the relationships between angles and side lengths of triangles. It is used in fields ranging from physics and engineering to astronomy, computer graphics, and music.

In a right triangle, sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, and tan θ = opposite/adjacent. Our calculator computes all six trig functions instantly from any angle in degrees or radians.

SOHCAHTOA is a memory aid: Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent.

The unit circle is a circle of radius 1 centered at the origin. Each point on it corresponds to (cos θ, sin θ) for some angle θ, making it the foundation of trig identities, periodic functions, and complex numbers.

Use the Pythagorean theorem (a² + b² = c²) to find missing sides, and inverse trig (arcsin, arccos, arctan) to find missing angles. The Right Triangle Solver above does this automatically.

Degrees divide a full circle into 360 parts; radians use the radius as a unit, so a full circle equals 2π radians. To convert: radians = degrees × π/180.

Use arcsin, arccos, or arctan when you know a ratio of sides and need to find the angle. For example, arctan(opposite/adjacent) returns the angle whose tangent equals that ratio.

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Solve Trigonometry Problems Instantly

Use our free Trigonometry Calculator to calculate trig functions, solve triangles, and explore mathematical relationships with confidence.

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