Triangle Solver & Trigonometry Studio
Master the relationship between sides and angles. Solve any triangle interactively and visualize SOH CAH TOA.
Right Triangle Explorer
Adjust the angle and hypotenuse to see how Opposite and Adjacent sides change correctly based on SOH CAH TOA.
The Language of Triangles
Triangles are the structural foundation of the universe. From the trusses holding up your roof to the graphics rendering this page, nature loves triangles.Trigonometry (from Greek trigonon "triangle" + metron "measure") is simply the study of how the **Sides** and **Angles** of a triangle are connected.
Right Triangles (SOH CAH TOA)
If your triangle has a 90° corner (a square corner), it's a "Right Triangle". This is the easiest playground. We use the famous mnemonic **SOH CAH TOA**:
- SOH: Sin(θ) = Opp / Hyp
- CAH: Cos(θ) = Adj / Hyp
- TOA: Tan(θ) = Opp / Adj
General Triangles (Oblique)
Most triangles in the real world are NOT right triangles. For these "Oblique Triangles", simple SOH CAH TOA doesn't work. We need heavier artillery:
Solving Strategies: A Field Guide
When you are faced with a "Solve the Triangle" problem, look at what you know:
- SSSSide-Side-Side: You know all three sides. Use the **Law of Cosines** first to find the largest angle (to avoid ambiguity), then Law of Sines for the rest.
- SASSide-Angle-Side: You know two sides and the angle between them. The **Law of Cosines** is perfect here to find the third missing side.
- ASAAngle-Side-Angle: You know two angles (which means you know all three, since they add to 180°). The **Law of Sines** is the fastest way home.
Real World Applications
Graphic Design
Every time you rotate an image in Photoshop or calculate a shadow in a 3D game, trigonometry matrices are doing the heavy lifting.
Construction
Carpenters use "rise over run" (Tangent) constantly to determine roof pitches and ramp angles to meet safety codes.
Surveying
To measure the width of a river without crossing it, surveyors measure a baseline on one side and two angles to a point on the other. Law of Sines does the rest.
Frequently Asked Questions
How do I find a missing side of a right triangle?
If you know one side and one angle, use SOH CAH TOA. For example, if you know the Hypotenuse (H) and an angle (θ), and want the Opposite side (O), use Sine: sin(θ) = O/H, so O = H * sin(θ). If you know two sides, use the Pythagorean Theorem: a² + b² = c².
When should I use the Law of Sines vs Law of Cosines?
Use the Law of Sines when you have a 'matching pair' (a side and its opposite angle) known. Use the Law of Cosines when you don't have a matching pair (e.g., you have SSS or SAS).
What is SOH CAH TOA?
It's a mnemonic to remember the primary trig ratios: Sine = Opposite / Hypotenuse, Cosine = Adjacent / Hypotenuse, Tangent = Opposite / Adjacent.
Can this tool solve non-right triangles?
Yes! The 'Triangle Solver' tab uses the General Laws of Trigonometry (Sines and Cosines) to solve Oblique triangles (triangles with no right angle) given enough information (like SSS, SAS, or ASA).
What is the 'Ambiguous Case' in trigonometry?
This happens in the SSA case (Side-Side-Angle) where two different triangles can be formed with the same given dimensions. This tool currently calculates the primary acute solution.
Why does tan(90°) return 'Undefined'?
Tangent is Sin/Cos. At 90 degrees, Cos is 0. Dividing by zero is impossible in standard arithmetic, so the value shoots to infinity (undefined).
How are trigonometric ratios used in real life?
They are essential in Engineering (bridge forces), Architecture (roof slopes), Computer Graphics (3D rotation), and Navigation (GPS triangulation).
What are the Reciprocal Identities?
They are the 'flip' of the main ratios: Cosecant (csc) = 1/sin, Secant (sec) = 1/cos, and Cotangent (cot) = 1/tan.
How do I convert Degrees to Radians?
Multiply your degrees by π/180. For example, 90° * (π/180) = π/2 radians.
Why do we use Radians in calculus?
Radians are the 'natural' unit for angles. Using them makes formulas for derivatives and integrals much simpler (e.g., derivative of sin(x) is cos(x) only if x is in radians).
What is the Pythagorean Identity?
It is the most famous identity: sin²θ + cos²θ = 1. It basically says the Pythagorean Theorem (a² + b² = c²) applies to the unit circle.
Can I use this for physics problems?
Absolutely. Resolving vectors into components (like splitting a velocity into x and y parts) is just applying SOH CAH TOA to a right triangle.