Trigonometry Calculator

Calculate trigonometric functions and solve right triangles with precise results

Two Sides
Enter two side lengths
One Angle + One Side
Enter an angle and a side
Area + One Side
Enter area and a side
C A B c a b α β γ = 90°

What is Trigonometry?

Trigonometry is a branch of mathematics that studies the relationships between angles and sides in triangles. The word comes from Greek: “trigōnon” (triangle) and “metron” (measure). This field of mathematics is essential for solving problems involving right triangles and calculating unknown angles or side lengths.

Basic Trigonometric Functions

The three primary trigonometric functions are sine (sin), cosine (cos), and tangent (tan). These functions relate the angles of a right triangle to the ratios of its sides:

sin(θ) = opposite / hypotenuse
cos(θ) = adjacent / hypotenuse
tan(θ) = opposite / adjacent

Reciprocal Functions

The reciprocal trigonometric functions provide alternative ways to express relationships:

cosecant (csc) = 1 / sin(θ)
secant (sec) = 1 / cos(θ)
cotangent (cot) = 1 / tan(θ)

How to Use This Trigonometry Calculator

Calculating Trigonometric Functions

To find trigonometric values for a specific angle:

  1. Enter your angle value in the input field
  2. Select the unit (degrees, radians, or gradians)
  3. Click “Calculate Trigonometric Values”
  4. View all six trigonometric function values with explanations

Solving Right Triangles

To find missing sides or angles in a right triangle:

  1. Select what information you know about the triangle
  2. Enter the known values with appropriate units
  3. Click “Solve Triangle”
  4. Review the complete triangle solution with all angles and sides

Real-World Applications

Architecture and Construction

Architects use trigonometry to calculate roof slopes, determine building heights, and create structurally sound designs. Construction workers apply these principles when installing staircases, ramps, and angled supports.

Navigation and GPS

Trigonometric calculations are fundamental to GPS systems, helping determine precise locations using satellite triangulation. Pilots and sailors use trigonometry for navigation and course plotting.

Physics and Engineering

Wave motion, oscillations, and periodic phenomena are described using trigonometric functions. Engineers use these calculations in designing bridges, analysing forces, and creating mechanical systems.

Computer Graphics

Game developers and animation software use trigonometry to create realistic movements, rotations, and 3D transformations. Screen coordinates and object positioning rely heavily on trigonometric calculations.

Common Trigonometric Values

Special Angles (Degrees)

sin(30°) = 1/2, cos(30°) = √3/2
sin(45°) = √2/2, cos(45°) = √2/2
sin(60°) = √3/2, cos(60°) = 1/2
sin(90°) = 1, cos(90°) = 0

Unit Circle Values

The unit circle provides a geometric representation of trigonometric functions. Key points help memorise common values and understand the cyclical nature of these functions.

Trigonometric Identities

sin²(θ) + cos²(θ) = 1
tan(θ) = sin(θ) / cos(θ)
sin(2θ) = 2sin(θ)cos(θ)
cos(2θ) = cos²(θ) – sin²(θ)

Frequently Asked Questions

What’s the difference between degrees and radians?
Degrees divide a circle into 360 equal parts, whilst radians are based on the radius of the circle. One complete revolution equals 360° or 2π radians. To convert: radians = degrees × π/180.
How do I remember SOHCAHTOA?
SOHCAHTOA represents: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. Popular memory aids include “Some Old Hippie Caught Another Hippie Tripping On Acid” or “Silly Old Harry Caught A Herring Trawling Off America”.
When are trigonometric functions undefined?
Functions become undefined when division by zero occurs. Tangent and secant are undefined when cosine equals zero (90°, 270°, etc.). Cotangent and cosecant are undefined when sine equals zero (0°, 180°, 360°, etc.).
Can I use trigonometry for non-right triangles?
Yes! The Law of Sines and Law of Cosines extend trigonometry to any triangle. These laws help solve triangles when you know different combinations of sides and angles, not just right triangles.
Why are my calculator results slightly different from hand calculations?
Small differences occur due to rounding in intermediate steps and the precision of decimal representations. Calculators typically use more decimal places internally, providing more accurate results than manual calculations with rounded values.
How accurate should my measurements be for practical applications?
Accuracy requirements depend on your application. Construction typically requires precision to the nearest centimetre or millimetre, whilst scientific calculations may need many decimal places. Consider the precision of your input measurements when interpreting results.

Tips for Mastering Trigonometry

Start with Right Triangles

Build confidence by practising with simple right triangles before tackling more complex problems. Master the basic ratios and their relationships first.

Visualise the Triangle

Always draw a diagram when solving trigonometry problems. Label the sides (opposite, adjacent, hypotenuse) and angles clearly to avoid confusion.

Check Your Answers

Use multiple methods to verify results. For example, if you find all three sides of a triangle, check that they satisfy the Pythagorean theorem: a² + b² = c².

Practice with Real Scenarios

Apply trigonometry to practical situations like measuring heights of buildings, calculating distances across rivers, or determining angles of elevation. Real-world context makes abstract concepts more meaningful.

Learn Common Patterns

Recognise special triangles (30-60-90 and 45-45-90) and memorise their side ratios. These patterns appear frequently and speed up problem-solving.

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