Trigonometry Calculator

Solve right-angled triangle problems using trigonometric functions

Enter the known values of a right triangle to find the unknown angles and sides. Supports sine, cosine, and tangent.

Examples

Click on an example to load it into the calculator.

Find Hypotenuse

findHypotenuse

Calculate the hypotenuse given the two other sides (a and b).

sideA: 3

sideB: 4

Find Angle A

findAngleA

Calculate Angle A given the opposite side (a) and adjacent side (b).

sideA: 5

sideB: 12

Find Opposite Side

findOpposite

Calculate the opposite side (a) given Angle A and the hypotenuse (c).

angleA: 30

sideC: 10

angleUnit: degrees

Real-World: Ladder Angle

findAdjacent

A 15m ladder leans against a wall, reaching 12m up. Find its angle with the ground.

sideA: 12

sideC: 15

calculationType: findAngleB

Other Titles
Understanding Trigonometry: A Comprehensive Guide
Explore the fundamentals of trigonometry, its real-world applications, and how to solve problems involving right-angled triangles.

What is Trigonometry? Foundations and Key Concepts

  • Trigonometry is the study of the relationships between angles and sides of triangles.
  • The primary functions are Sine (sin), Cosine (cos), and Tangent (tan).
  • SOHCAHTOA is a mnemonic to remember the trigonometric ratios.
Trigonometry is a branch of mathematics that deals with the relationships between the angles and side lengths of triangles. While it applies to all triangles, it is most commonly introduced with right-angled triangles. The core of trigonometry lies in three primary functions: sine, cosine, and tangent.
The Trigonometric Ratios (SOHCAHTOA)

For an angle θ in a right-angled triangle:

  • Sine (sin θ) = Length of Opposite Side / Length of Hypotenuse
  • Cosine (cos θ) = Length of Adjacent Side / Length of Hypotenuse
  • Tangent (tan θ) = Length of Opposite Side / Length of Adjacent Side
The mnemonic SOHCAHTOA helps remember these ratios: Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, and Tangent is Opposite over Adjacent.

Basic Trigonometric Relationships

  • In a 3-4-5 triangle, for the angle opposite the side of length 3, sin(θ) = 3/5.
  • If Angle A is 30°, cos(30°) ≈ 0.866.
  • If the opposite side is 5 and adjacent is 12, tan(θ) = 5/12.

Step-by-Step Guide to Using the Trigonometry Calculator

  • Select the value you want to find.
  • Input the known measurements of the triangle.
  • Choose your preferred angle unit (Degrees or Radians).
1. Choose Your Goal
Start by selecting what you want to calculate from the 'What to Find' dropdown menu. You can choose to find one of the angles (A or B) or one of the sides (opposite, adjacent, or hypotenuse).
2. Enter Known Values
The calculator will dynamically show the required input fields based on your selection. For example, to find the hypotenuse, you'll need to enter the lengths of the other two sides. To find an angle, you might need two sides.
3. Select Angle Units
If your calculation involves angles, ensure you select the correct unit: Degrees or Radians. This is crucial for accurate results.
4. Interpret the Results
After clicking 'Calculate', the tool will display all properties of the triangle, including the calculated value, other unknown sides/angles, area, and perimeter.

Practical Calculation Steps

  • To find Hypotenuse: Select 'Side Hypotenuse', enter Side A = 3, Side B = 4. Result: Hypotenuse = 5.
  • To find Angle A: Select 'Angle A', enter Side A (Opposite) = 5, Side C (Hypotenuse) = 10. Result: Angle A = 30°.

Real-World Applications of Trigonometry

  • Architecture and Engineering: Designing structures and calculating forces.
  • Navigation and Astronomy: Pinpointing locations and mapping celestial bodies.
  • Video Games and Computer Graphics: Rendering 3D environments and character movements.
Architecture and Construction
Trigonometry is fundamental in architecture to calculate roof slopes, ground gradients, and structural load angles. It ensures buildings are stable, safe, and built to specification.
Navigation and GPS
Trigonometry is used in GPS systems to determine a precise location. By receiving signals from multiple satellites, a GPS receiver uses trigonometric principles (specifically triangulation) to calculate its exact position on Earth.
Physics and Engineering
In physics, trigonometry is used to analyze vectors, model wave mechanics (like sound and light), and calculate projectile motion. Engineers use it to design circuits, analyze forces on bridges, and in mechanical engineering.

Trigonometry in Action

  • Calculating the height of a flagpole by measuring the distance from its base and the angle of elevation to the top.
  • Determining the path of a satellite as it orbits the Earth.
  • Creating realistic lighting and shadow effects in a video game.

Key Formulas in Trigonometry

  • The Pythagorean Theorem is central to right-angled triangles.
  • Inverse trigonometric functions allow calculating angles from side ratios.
  • The sum of angles in any triangle is always 180°.
Pythagorean Theorem
For a right-angled triangle with sides a, b, and hypotenuse c, the theorem states: a² + b² = c². This is essential for finding a side when two other sides are known.
Inverse Trigonometric Functions

When you know the ratio of the sides but need to find the angle, you use inverse functions:

  • Arcsin (sin⁻¹) finds the angle whose sine is a given number.
  • Arccos (cos⁻¹) finds the angle whose cosine is a given number.
  • Arctan (tan⁻¹) finds the angle whose tangent is a given number.
Sum of Angles
For any triangle, the sum of its three interior angles is 180 degrees (or π radians). In a right-angled triangle, since one angle is 90°, the other two acute angles must sum to 90°.

Core Formulas

  • If a=8, b=15, then c = √(8² + 15²) = √289 = 17.
  • If sin(A) = 0.5, then A = arcsin(0.5) = 30°.
  • If Angle A = 40°, then Angle B = 90° - 40° = 50°.

Common Misconceptions and Best Practices

  • Ensure your calculator is in the correct mode (Degrees or Radians).
  • The hypotenuse is always the longest side, opposite the right angle.
  • SOHCAHTOA only applies to right-angled triangles.
Degrees vs. Radians
One of the most common errors is having the calculator in the wrong mode. Calculating sin(30) in radians mode will give a completely different answer than in degrees mode. Always double-check your angle unit.
Identifying Sides Correctly
Correctly identifying the 'opposite' and 'adjacent' sides is crucial. These are relative to the angle you are working with. The side opposite one acute angle is adjacent to the other.
Scope of SOHCAHTOA
The SOHCAHTOA rules are a simplification for right-angled triangles only. For other triangles (oblique triangles), you must use the Law of Sines or the Law of Cosines to solve for unknown sides and angles.

Tips for Accuracy

  • Check Mode: If sin(A) should be 0.5 and you get -0.988, you're likely in Radians mode.
  • Hypotenuse Check: If you calculate a side that is longer than the hypotenuse, re-check your inputs and formulas.
  • Triangle Type: For non-right triangles, use a Law of Sines/Cosines calculator.