Trigonometry Calculator

Solve any triangle and explore trigonometric identities

Triangle Solver

Enter any 3 values (at least one side) to find the rest.

Solution

Enter 3 values to solve...

51 Trigonometric Identities

Reciprocal Identities

sin θ = 1/csc θ

cos θ = 1/sec θ

tan θ = 1/cot θ

csc θ = 1/sin θ

sec θ = 1/cos θ

cot θ = 1/tan θ

Quotient Identities

tan θ = sin θ/cos θ

cot θ = cos θ/sin θ

Pythagorean Identities

sin²θ + cos²θ = 1

1 + tan²θ = sec²θ

1 + cot²θ = csc²θ

Even/Odd Identities

sin(-θ) = -sin θ

cos(-θ) = cos θ

tan(-θ) = -tan θ

csc(-θ) = -csc θ

sec(-θ) = sec θ

cot(-θ) = -cot θ

Co-Function Identities

sin(π/2 - θ) = cos θ

cos(π/2 - θ) = sin θ

tan(π/2 - θ) = cot θ

csc(π/2 - θ) = sec θ

sec(π/2 - θ) = csc θ

cot(π/2 - θ) = tan θ

Sum & Difference Formulas

sin(α ± β) = sin α cos β ± cos α sin β

cos(α ± β) = cos α cos β ∓ sin α sin β

tan(α ± β) = (tan α ± tan β) / (1 ∓ tan α tan β)

Double Angle Formulas

sin(2θ) = 2sin θ cos θ

cos(2θ) = cos²θ - sin²θ

cos(2θ) = 2cos²θ - 1

cos(2θ) = 1 - 2sin²θ

tan(2θ) = 2tan θ / (1 - tan²θ)

Half Angle Formulas

sin(θ/2) = ±√[(1 - cos θ)/2]

cos(θ/2) = ±√[(1 + cos θ)/2]

tan(θ/2) = ±√[(1 - cos θ)/(1 + cos θ)]

Sum to Product Formulas

sin α + sin β = 2sin((α+β)/2)cos((α-β)/2)

sin α - sin β = 2cos((α+β)/2)sin((α-β)/2)

cos α + cos β = 2cos((α+β)/2)cos((α-β)/2)

cos α - cos β = -2sin((α+β)/2)sin((α-β)/2)

Product to Sum Formulas

sin α sin β = ½[cos(α-β) - cos(α+β)]

cos α cos β = ½[cos(α-β) + cos(α+β)]

sin α cos β = ½[sin(α+β) + sin(α-β)]

What is it used for?

The Trigonometry Calculator solves for missing sides and angles in right and oblique triangles.

Benefits & Tips

  • Law of Sines/Cosines: Automatically applies the correct law
  • Identities Reference: Quick access to all 51 standard formulas
  • Visual Aid: Helps visualize triangle properties
  • Tip: Ensure triangle inequality (a+b>c) holds true

The Ultimate Guide to Trigonometry & Triangle Solvers

Unlocking the relationships between side lengths and angles in geometry.

What is Trigonometry?

Trigonometry (from Greek trigonon, "triangle" and metron, "measure") is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. Today, it is essential in fields ranging from architecture and engineering to physics, astronomy, and digital imaging.

The core of trigonometry lies in the six fundamental trigonometric functions: Sine ($\sin$), Cosine ($\cos$), Tangent ($\tan$), Cosecant ($\csc$), Secant ($\sec$), and Cotangent ($\cot$). These functions map an angle of a right-angled triangle to the ratio of two of its side lengths.

Solving Triangles: The Laws of Sines and Cosines

Our Triangle Solver is programmed to compute any missing variables (sides or angles) of a triangle as long as three parameters are known (with at least one being a side length). It achieves this by automatically applying two critical theorems:

The Law of Sines

Used when you know two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA). It states that the ratio of a side to the sine of its opposite angle is constant for all three sides.

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$
The Law of Cosines

A generalized form of the Pythagorean theorem. Used when you know two sides and the included angle (SAS) or all three sides (SSS).

$c^2 = a^2 + b^2 - 2ab \cdot \cos(C)$

Why are Trigonometric Identities Important?

The 51 trigonometric identities listed above are equality statements that hold true for all values of the variables. But why do we need them? In higher-level calculus, physics, and engineering, you will often encounter complex wave equations or differential formulas that are nearly impossible to solve in their raw form.

Identities like the Pythagorean Identity ($\sin^2\theta + \cos^2\theta = 1$) or the Double Angle Formulas allow mathematicians to manipulate and simplify these expressions into solvable formats. They are the algebraic rules of the trigonometric world.

Real-World Use Cases

  • Architecture & Construction: Calculating structural load vectors, determining roof pitches, and ensuring structural stability.
  • Music Theory & Acoustics: Sound travels in waves. The modeling of sound waves, frequency tuning, and digital audio compression (like MP3 algorithms) rely entirely on Fourier transforms, which are built on sine and cosine functions.
  • Navigation (Marine & Aviation): Calculating distances between coordinates using the spherical law of cosines and triangulating positions via GPS satellites.