Trigonometric Identity Verifier
Verify and evaluate key trigonometric identities — Pythagorean, double-angle sine, and double-angle cosine — for any angle in degrees or radians. Great for checking algebra and studying trig transformations.
About this calculator
Trigonometric identities are equations that hold true for all angle values. The Pythagorean identity states sin²(θ) + cos²(θ) = 1, a fundamental result derived from the unit circle definition of sine and cosine. The double-angle identities express trig functions of 2θ in terms of θ: sin(2θ) = 2·sin(θ)·cos(θ) and cos(2θ) = cos²(θ) − sin²(θ). These identities are used to simplify expressions, solve equations, and prove further results in calculus and signal processing. When working in degrees, angles are first converted to radians via θ_rad = θ × π/180 before being passed to the sine and cosine functions. The calculator evaluates each identity numerically, allowing you to confirm that both sides of the equation produce the same value for a given angle.
How to use
Verify the double-angle sine identity at θ = 30°. The identity states sin(2θ) = 2·sin(θ)·cos(θ). Direct calculation: sin(60°) = 0.8660. Via the identity: 2 × sin(30°) × cos(30°) = 2 × 0.5 × 0.8660 = 0.8660. Enter 30 in the angle field, select 'doubleAngleSin' as the identity, choose 'degrees' as the input unit, and set precision to 4. The calculator returns 0.8660, confirming both sides match. Try the Pythagorean identity with any angle — it always returns 1.0000.
Frequently asked questions
Why does the Pythagorean identity sin²(θ) + cos²(θ) always equal 1 for any angle?
The identity follows directly from the unit circle, where any point on the circle satisfies x² + y² = 1. By definition, cos(θ) = x and sin(θ) = y for an angle θ measured from the positive x-axis. Substituting gives cos²(θ) + sin²(θ) = 1. This holds for every real angle without exception, making it one of the most useful identities in mathematics. It can also be derived from the Pythagorean theorem applied to a right triangle inscribed in the unit circle.
How are double-angle formulas used in real-world applications?
Double-angle formulas simplify expressions that arise naturally in physics, engineering, and signal processing. In mechanics, the range of a projectile involves sin(2θ), which is rewritten as 2·sin(θ)·cos(θ) to analyze the launch angle independently. In electronics, multiplying two sine waves of the same frequency produces terms involving double-angle cosines — a key step in amplitude modulation (AM) and mixing. In calculus, double-angle identities help evaluate integrals of sin²(x) and cos²(x) by converting them into single-angle terms.
What is the difference between an identity and an equation in trigonometry?
A trigonometric equation is true only for specific values of the angle — for example, sin(θ) = 0.5 is satisfied only when θ = 30°, 150°, and their periodic equivalents. A trigonometric identity, by contrast, is true for all values of the variable within its domain. Verifying an identity numerically at one angle confirms it there but does not prove it universally; a proper proof requires algebraic manipulation valid for all angles. This calculator confirms the numerical agreement at a chosen angle, which is useful for checking algebra but not a formal proof.