Trigonometry Calculator
Our free trigonometry calculator helps you compute values of key trigonometric functions based on angles or triangle sides. Whether you're working on geometry problems, physics applications, or engineering calculations, this tool provides accurate sine, cosine, tangent and other trig functions with just a few clicks. Perfect for students, teachers, and professionals who need quick trigonometry solutions.
Important Notes:
- This calculator provides accurate values for all major trigonometric functions.
- For angles in degrees, the range 0° to 360° covers a full rotation.
- For angles in radians, the range 0 to 2π (approximately 6.28) covers a full rotation.
- Some trigonometric functions have undefined values at certain angles (e.g., tangent at 90°).
- The calculator uses the standard right triangle definition of trigonometric functions.
- Inverse trigonometric functions may have restricted output ranges.
Understanding Trigonometry: A Complete Guide
Trigonometry is a branch of mathematics that studies the relationships between the sides and angles of triangles. Derived from Greek words "trigōnon" (triangle) and "metron" (measure), trigonometry has applications across various fields including physics, engineering, astronomy, and architecture.
What is Trigonometry?
Trigonometry primarily focuses on right triangles (triangles with one 90-degree angle). The fundamental trigonometric functions—sine, cosine, and tangent—express the ratios between different sides of a right triangle relative to its angles. These functions serve as the foundation for countless mathematical concepts and real-world applications.
Basic Trigonometric Functions
The six main trigonometric functions are:
- Sine (sin): The ratio of the opposite side to the hypotenuse
- Cosine (cos): The ratio of the adjacent side to the hypotenuse
- Tangent (tan): The ratio of the opposite side to the adjacent side
- Cosecant (csc): The reciprocal of sine
- Secant (sec): The reciprocal of cosine
- Cotangent (cot): The reciprocal of tangent
Applications of Trigonometry
Trigonometry has widespread applications across various disciplines:
- Physics: Analyzing forces, waves, and oscillations
- Engineering: Designing structures, electrical systems, and mechanical components
- Navigation: Calculating distances and positions in aviation and maritime travel
- Architecture: Planning and designing buildings with specific angles and dimensions
- Astronomy: Measuring celestial distances and mapping planetary movements
- Computer Graphics: Creating animations, games, and visual simulations
Frequently Asked Questions
What are the basic trigonometric functions?
The three basic trigonometric functions are sine (sin), cosine (cos), and tangent (tan). These are defined using the sides of a right triangle: sine is the ratio of the opposite side to the hypotenuse, cosine is the ratio of the adjacent side to the hypotenuse, and tangent is the ratio of the opposite side to the adjacent side. The other three functions—cosecant (csc), secant (sec), and cotangent (cot)—are reciprocals of sine, cosine, and tangent, respectively.
How do I calculate sine, cosine, and tangent?
To calculate sine, cosine, and tangent for an angle in a right triangle:
- sine (sin) = opposite side ÷ hypotenuse
- cosine (cos) = adjacent side ÷ hypotenuse
- tangent (tan) = opposite side ÷ adjacent side
You can use our calculator to compute these values automatically by entering either the angle or the triangle sides.
What's the difference between radians and degrees?
Degrees and radians are two different units for measuring angles:
- Degrees: A full circle is 360 degrees (360°). This is the more commonly used unit in everyday applications.
- Radians: A full circle is 2π radians (approximately 6.28). Radians are often preferred in advanced mathematics and physics because they simplify many formulas.
To convert between them: 1 radian ≈ 57.3 degrees, and 1 degree ≈ 0.01745 radians.
Can I calculate trig functions without an angle?
Yes, you can calculate trigonometric functions using the sides of a right triangle without knowing the angle. If you know any two sides of a right triangle, you can calculate the ratios that define the trigonometric functions. For example, if you know the opposite side (3) and hypotenuse (5), you can calculate sine as 3/5 = 0.6. Our calculator provides a "Triangle Side Ratios" mode specifically for this purpose.
What are inverse trigonometric functions used for?
Inverse trigonometric functions (arcsin, arccos, arctan) are used to find the angle when you know the trigonometric ratio. For example, if you know that sin(θ) = 0.5, you can use arcsin(0.5) to find that θ = 30 degrees. These functions are particularly useful in:
- Finding angles in triangles when you know the sides
- Navigational calculations
- Physics problems involving vector components
- Engineering applications requiring angular measurements