Trig Identities Calculator
Calculate all trig functions and verify identities
How to Use This Calculator
Enter the Angle
Input any angle value in degrees or radians.
Select Unit
Choose whether your angle is in degrees or radians.
Calculate
Click "Calculate All Identities" to see all six trigonometric functions and verify fundamental identities.
Formula
Fundamental Trigonometric Identities:
Pythagorean Identities:
- sin²(θ) + cos²(θ) = 1
- 1 + tan²(θ) = sec²(θ)
- 1 + cot²(θ) = csc²(θ)
Quotient Identities:
- tan(θ) = sin(θ) / cos(θ)
- cot(θ) = cos(θ) / sin(θ)
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)
About Trig Identities Calculator
The Trig Identities Calculator calculates all six trigonometric functions (sin, cos, tan, csc, sec, cot) for any angle and verifies fundamental trigonometric identities. These identities are essential relationships that hold for all angles and form the foundation of trigonometry.
What are Trigonometric Identities?
Trigonometric identities are equations that are true for all valid values of the angle. The three main categories are Pythagorean identities (relating sin² and cos²), quotient identities (relating tan to sin/cos), and reciprocal identities (relating each function to its reciprocal).
When to Use This Calculator
- Verification: Verify that identities hold for specific angles
- Learning: Understand relationships between trigonometric functions
- Problem Solving: Use identities to simplify expressions
- Checking Work: Verify manual calculations
Why Use Our Calculator?
- ✅ All Functions: Calculates all six trigonometric functions at once
- ✅ Identity Verification: Automatically verifies fundamental identities
- ✅ Visual Feedback: Shows checkmarks when identities hold
- ✅ 100% Free: No registration required
- ✅ Educational: Helps understand trigonometric relationships
Frequently Asked Questions
Why do trigonometric identities exist?
Trigonometric identities exist because of the geometric relationships in right triangles and the unit circle. They're fundamental properties that always hold true.
What's the most important trigonometric identity?
The Pythagorean identity sin²(θ) + cos²(θ) = 1 is considered the most fundamental, as it comes directly from the unit circle and is the basis for many other identities.
How are identities different from equations?
Identities are true for all valid values of the variable, while equations are only true for specific values. For example, sin²(θ) + cos²(θ) = 1 is an identity (true for all θ), while sin(θ) = 0.5 is an equation (only true for specific angles).