Trigonometry Calculator

Trigonometry Calculator: Complete Guide to Trigonometric Functions

What is Trigonometry?

Trigonometry is a branch of mathematics that studies relationships between the angles and sides of triangles. The word "trigonometry" comes from Greek words meaning "triangle measurement." This field is essential in mathematics, physics, engineering, and many other scientific disciplines.

Basic Trigonometric Functions

Primary Functions

The three primary trigonometric functions are defined using a right triangle:

• Sine (sin θ) = opposite side / hypotenuse
• Cosine (cos θ) = adjacent side / hypotenuse
• Tangent (tan θ) = opposite side / adjacent side

Reciprocal Functions

The three reciprocal trigonometric functions are:

• Cosecant (csc θ) = 1/sin θ = hypotenuse / opposite side
• Secant (sec θ) = 1/cos θ = hypotenuse / adjacent side
• Cotangent (cot θ) = 1/tan θ = adjacent side / opposite side

Right Triangle Trigonometry

The Pythagorean Theorem

In a right triangle with sides a, b, and hypotenuse c:

a² + b² = c²

Common Right Triangle Ratios

Angle	sin θ	cos θ	tan θ
30°	1/2	√3/2	1/√3
45°	√2/2	√2/2	1
60°	√3/2	1/2	√3

Unit Circle and Radian Measure

Degrees vs. Radians

Angles can be measured in degrees or radians:

• 360° = 2π radians
• 180° = π radians
• 90° = π/2 radians
• To convert: radians = degrees × π/180
• To convert: degrees = radians × 180/π

Unit Circle Values

The unit circle (radius = 1) provides trigonometric values for any angle. Key points on the unit circle correspond to common angles and their trigonometric ratios.

Inverse Trigonometric Functions

Arcsine, Arccosine, and Arctangent

Inverse trigonometric functions find angles when given trigonometric ratios:

• arcsine (sin⁻¹): finds angle whose sine is the given value
• arccosine (cos⁻¹): finds angle whose cosine is the given value
• arctangent (tan⁻¹): finds angle whose tangent is the given value

Trigonometric Identities

Fundamental Identities

• sin²θ + cos²θ = 1 (Pythagorean identity)
• tan θ = sin θ / cos θ
• 1 + tan²θ = sec²θ
• 1 + cot²θ = csc²θ

Angle Sum and Difference Formulas

• sin(A ± B) = sin A cos B ± cos A sin B
• cos(A ± B) = cos A cos B ∓ sin A sin B
• tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B)

Applications of Trigonometry

Navigation and Surveying

Trigonometry is used to calculate distances, heights, and directions in navigation, GPS systems, and land surveying.

Engineering and Architecture

Engineers use trigonometry for structural analysis, wave mechanics, signal processing, and architectural design.

Physics and Astronomy

Trigonometric functions model periodic phenomena like waves, oscillations, and celestial movements.

Computer Graphics

3D graphics, rotations, transformations, and animations rely heavily on trigonometric calculations.

Problem-Solving Strategies

• Draw a clear diagram when solving triangle problems
• Identify known and unknown values
• Choose appropriate trigonometric functions
• Use the Pythagorean theorem when applicable
• Check answers for reasonableness
• Convert between degrees and radians as needed

Calculator Features

Our trigonometry calculator provides comprehensive tools for all trigonometric calculations. Calculate function values, solve right triangles, evaluate complex expressions, and convert between angle units with precision and ease. Perfect for students, engineers, and professionals working with trigonometric problems.