Trig Identity Calculator
Trigonometric identities are equations involving trig functions that hold true for all valid angle values. They are the algebraic backbone of trigonometry, used to simplify expressions, solve equations, derive new formulas and build the calculus of trig functions. The most fundamental are the three Pythagorean identities, which follow directly from the unit circle definition of sine and cosine. Then there are the double angle identities, which express sin(2 theta) and cos(2 theta) in terms of sin(theta) and cos(theta). The sum identities express sin(A + B) and cos(A + B) in expanded form. The half-angle identities give sin(theta/2) and cos(theta/2). All of these are exact algebraic relationships, not approximations. This calculator lets you enter any angle in degrees and see all six trig values at that angle (sine, cosine, tangent, cotangent, secant, cosecant) along with a numerical verification of the major identities. Each identity shows you the left-hand side value, the right-hand side value and a pass or fail check to nine decimal places. At 45 degrees for instance, sin squared(45) + cos squared(45) = 0.5 + 0.5 = 1.0000 exactly, passing the Pythagorean identity check. The tool suits students studying for exams, teachers building class demonstrations, and anyone who wants to explore how the identities behave across different angles.
| Identity | LHS | RHS | Check |
|---|---|---|---|
| sin²θ + cos²θ = 1 | 1.0000 | 1.0000 | PASS |
| 1 + tan²θ = sec²θ | 2.0000 | 2.0000 | PASS |
| 1 + cot²θ = csc²θ | 2.0000 | 2.0000 | PASS |
| sin(2θ) = 2sinθcosθ | 1.0000 | 1.0000 | PASS |
| cos(2θ) = cos²θ - sin²θ | 0.0000 | 0.0000 | PASS |
| cos(2θ) = 1 - 2sin²θ | 0.0000 | 0.0000 | PASS |
Pass means LHS and RHS agree to within 1e-9. All identities hold for any angle where the functions are defined.
How it works
The calculator converts the angle to radians and computes sine and cosine using JavaScript's built-in Math.sin and Math.cos. Tangent, cotangent, secant and cosecant are derived as sin/cos, cos/sin, 1/cos and 1/sin. Each identity is then evaluated numerically on both sides and compared; if they agree within 1e-9, the identity passes. The Pythagorean identities are: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = csc²θ. The double angle identities are: sin(2θ) = 2sinθcosθ, cos(2θ) = cos²θ - sin²θ, cos(2θ) = 1 - 2sin²θ.
Worked example
At theta = 45 degrees: sin(45) = cos(45) = 0.7071. sin²(45) + cos²(45) = 0.5 + 0.5 = 1.0000. sin(90) = 2 × 0.7071 × 0.7071 = 1.0000. cos(90) = cos²(45) - sin²(45) = 0.5 - 0.5 = 0.0000. All identities pass at this angle, matching the defaults pre-filled in the calculator above.
Frequently asked questions
What are the three Pythagorean trig identities?
The three Pythagorean identities are: sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; and 1 + cot²θ = csc²θ. All three follow from the unit circle definition of sine and cosine and apply for all angles where the functions are defined.
What is the double angle identity for sin(2θ)?
The double angle identity for sine is sin(2θ) = 2 sinθ cosθ. For cosine there are three equivalent forms: cos(2θ) = cos²θ - sin²θ = 2 cos²θ - 1 = 1 - 2 sin²θ. All three cosine forms are equivalent and can be derived from the Pythagorean identity.
How do you verify a trig identity numerically?
To verify an identity numerically, substitute a specific angle and check that the left-hand side equals the right-hand side to many decimal places. For example, for theta = 45 degrees: sin²(45) + cos²(45) = 0.5 + 0.5 = 1. While numerical checking confirms an identity for that angle, a formal algebraic proof is needed to establish it for all angles.
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